<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_430><Z_259><Z_102><Z_405><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_417><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_271><Z_97><Z_504><Z_213><Z_98><ANSWER>

Question: Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_305><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

Question:
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_358><Z_457><Z_234><Z_387><Z_200><Z_122><Z_287><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_61><Z_484><Z_43><Z_371><Z_503><Z_400><Z_97><ANSWER>

Question: A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_5><Z_409><Z_452><Z_351><Z_270><Z_131><Z_463><Z_203><Z_339><Z_83><Z_401><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_106><Z_181><Z_228><Z_110><Z_38><Z_474><Z_426><Z_432><Z_38><Z_45><Z_438><ANSWER>

Question: Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_66><Z_187><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_134><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_185><Z_431><Z_91><ANSWER>

Question: A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_195><Z_172><Z_187><Z_444><Z_94><Z_405><Z_79><Z_386><Z_461><Z_232><Z_259><Z_189><Z_477><Z_465><Z_491><Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|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
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_482><Z_466><Z_254><Z_415><Z_294><Z_409><Z_366><Z_78><Z_194><Z_187><Z_383><Z_187><Z_62><Z_160><Z_290><Z_406><ANSWER>

Question: Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_176><Z_254><Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_463><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_475><Z_118><Z_235><ANSWER>

Question: Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_66><Z_449><Z_228><Z_380><Z_303><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_29><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_148><Z_60><Z_433><Z_185><Z_349><ANSWER>

Question: A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).<|im_end|>
<|im_start|>assistant
<Z_195><Z_121><Z_155><Z_318><Z_376><Z_269><Z_418><Z_124><Z_105><Z_479><Z_333><Z_385><Z_509><Z_295><Z_345><Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).<|im_end|>
<|im_start|>assistant
<Z_255><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).<|im_end|>
<|im_start|>assistant
<Z_407><Z_201><Z_495><Z_131><Z_305><Z_415><Z_15><Z_75><Z_175><Z_127><Z_119><Z_219><Z_359><Z_207><Z_25><Z_348><ANSWER>

Question:
Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A puppy, two cats, and a bird together weigh 34 pounds. The puppy and the larger cat together weigh exactly three times as much as the bird, and the puppy and the smaller cat together weigh exactly twice as much as the bird. How many pounds does the puppy weigh?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A puppy, two cats, and a bird together weigh 34 pounds. The puppy and the larger cat together weigh exactly three times as much as the bird, and the puppy and the smaller cat together weigh exactly twice as much as the bird. How many pounds does the puppy weigh?<|im_end|>
<|im_start|>assistant
<Z_66><Z_415><Z_199><Z_81><Z_495><Z_435><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A puppy, two cats, and a bird together weigh 34 pounds. The puppy and the larger cat together weigh exactly three times as much as the bird, and the puppy and the smaller cat together weigh exactly twice as much as the bird. How many pounds does the puppy weigh?<|im_end|>
<|im_start|>assistant
<Z_290><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A puppy, two cats, and a bird together weigh 34 pounds. The puppy and the larger cat together weigh exactly three times as much as the bird, and the puppy and the smaller cat together weigh exactly twice as much as the bird. How many pounds does the puppy weigh?<|im_end|>
<|im_start|>assistant
<Z_173><Z_193><Z_423><Z_244><Z_366><Z_120><ANSWER>

Question: A puppy, two cats, and a bird together weigh 34 pounds. The puppy and the larger cat together weigh exactly three times as much as the bird, and the puppy and the smaller cat together weigh exactly twice as much as the bird. How many pounds does the puppy weigh?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the positive divisors of 36.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the positive divisors of 36.<|im_end|>
<|im_start|>assistant
<Z_5><Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the positive divisors of 36.<|im_end|>
<|im_start|>assistant
<Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the positive divisors of 36.<|im_end|>
<|im_start|>assistant
<Z_311><Z_407><ANSWER>

Question: Find the sum of the positive divisors of 36.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.<|im_end|>
<|im_start|>assistant
<Z_5><Z_461><Z_392><Z_458><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.<|im_end|>
<|im_start|>assistant
<Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.<|im_end|>
<|im_start|>assistant
<Z_9><Z_442><Z_302><Z_242><ANSWER>

Question:
Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.<|im_end|>
<|im_start|>assistant
<Z_5><Z_294><Z_407><Z_503><Z_229><Z_500><Z_193><Z_183><Z_251><Z_447><Z_98><Z_97><Z_251><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.<|im_end|>
<|im_start|>assistant
<Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.<|im_end|>
<|im_start|>assistant
<Z_495><Z_86><Z_511><Z_269><Z_191><Z_403><Z_480><Z_134><Z_131><Z_154><Z_26><Z_71><Z_439><ANSWER>

Question: Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1+a_2=10$, $a_4-a_3=2$.
(I) Find the general formula for $\{a_n\}$.
(II) Suppose a geometric sequence $\{b_n\}$ satisfies $b_2=a_3$, $b_3=a_7$, what is the position of the term in $\{a_n\}$ that equals $b_6$?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1+a_2=10$, $a_4-a_3=2$.
(I) Find the general formula for $\{a_n\}$.
(II) Suppose a geometric sequence $\{b_n\}$ satisfies $b_2=a_3$, $b_3=a_7$, what is the position of the term in $\{a_n\}$ that equals $b_6$?<|im_end|>
<|im_start|>assistant
<Z_5><Z_503><Z_473><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1+a_2=10$, $a_4-a_3=2$.
(I) Find the general formula for $\{a_n\}$.
(II) Suppose a geometric sequence $\{b_n\}$ satisfies $b_2=a_3$, $b_3=a_7$, what is the position of the term in $\{a_n\}$ that equals $b_6$?<|im_end|>
<|im_start|>assistant
<Z_362><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1+a_2=10$, $a_4-a_3=2$.
(I) Find the general formula for $\{a_n\}$.
(II) Suppose a geometric sequence $\{b_n\}$ satisfies $b_2=a_3$, $b_3=a_7$, what is the position of the term in $\{a_n\}$ that equals $b_6$?<|im_end|>
<|im_start|>assistant
<Z_423><Z_258><Z_120><Z_450><ANSWER>

Question:
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1+a_2=10$, $a_4-a_3=2$.
(I) Find the general formula for $\{a_n\}$.
(II) Suppose a geometric sequence $\{b_n\}$ satisfies $b_2=a_3$, $b_3=a_7$, what is the position of the term in $\{a_n\}$ that equals $b_6$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
amit and ananthu can do a work in 15 days and 30 days respectively . amit started the work and left after 3 days . ananthu took over and completed the work . in how many days was the total work completed ?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
amit and ananthu can do a work in 15 days and 30 days respectively . amit started the work and left after 3 days . ananthu took over and completed the work . in how many days was the total work completed ?<|im_end|>
<|im_start|>assistant
<Z_66><Z_139><Z_105><Z_409><Z_259><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
amit and ananthu can do a work in 15 days and 30 days respectively . amit started the work and left after 3 days . ananthu took over and completed the work . in how many days was the total work completed ?<|im_end|>
<|im_start|>assistant
<Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
amit and ananthu can do a work in 15 days and 30 days respectively . amit started the work and left after 3 days . ananthu took over and completed the work . in how many days was the total work completed ?<|im_end|>
<|im_start|>assistant
<Z_241><Z_295><Z_0><Z_42><Z_139><Z_480><ANSWER>

Question: amit and ananthu can do a work in 15 days and 30 days respectively . amit started the work and left after 3 days . ananthu took over and completed the work . in how many days was the total work completed ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).<|im_end|>
<|im_start|>assistant
<Z_195><Z_345><Z_510><Z_435><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).<|im_end|>
<|im_start|>assistant
<Z_233><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).<|im_end|>
<|im_start|>assistant
<Z_358><Z_181><Z_7><Z_85><ANSWER>

Question: Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?<|im_end|>
<|im_start|>assistant
<Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?<|im_end|>
<|im_start|>assistant
<Z_13><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?<|im_end|>
<|im_start|>assistant
<Z_368><ANSWER>

Question: Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $m$ for which $\mathop{\text{lcm}}[m, 15]=45$?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $m$ for which $\mathop{\text{lcm}}[m, 15]=45$?<|im_end|>
<|im_start|>assistant
<Z_66><Z_171><Z_148><Z_340><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $m$ for which $\mathop{\text{lcm}}[m, 15]=45$?<|im_end|>
<|im_start|>assistant
<Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $m$ for which $\mathop{\text{lcm}}[m, 15]=45$?<|im_end|>
<|im_start|>assistant
<Z_101><Z_300><Z_218><Z_295><ANSWER>

Question: What is the sum of all positive integers $m$ for which $\mathop{\text{lcm}}[m, 15]=45$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of the polynomial \( p(x) = x^3 - 8x^2 + 17x - 10 \).<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of the polynomial \( p(x) = x^3 - 8x^2 + 17x - 10 \).<|im_end|>
<|im_start|>assistant
<Z_195><Z_311><Z_321><Z_112><Z_407><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of the polynomial \( p(x) = x^3 - 8x^2 + 17x - 10 \).<|im_end|>
<|im_start|>assistant
<Z_416><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of the polynomial \( p(x) = x^3 - 8x^2 + 17x - 10 \).<|im_end|>
<|im_start|>assistant
<Z_110><Z_364><Z_58><Z_255><Z_3><ANSWER>

Question: Find the sum of the rational roots of the polynomial \( p(x) = x^3 - 8x^2 + 17x - 10 \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If circle $O_1: x^2+y^2=5$ and circle $O_2: (x-m)^2+y^2=20$ (where $m \in \mathbb{R}$) intersect at points A and B, and the tangents to the circles at point A are perpendicular to each other, then the length of segment AB is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If circle $O_1: x^2+y^2=5$ and circle $O_2: (x-m)^2+y^2=20$ (where $m \in \mathbb{R}$) intersect at points A and B, and the tangents to the circles at point A are perpendicular to each other, then the length of segment AB is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_195><Z_304><Z_199><Z_311><Z_80><Z_223><Z_476><Z_376><Z_126><Z_503><Z_260><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If circle $O_1: x^2+y^2=5$ and circle $O_2: (x-m)^2+y^2=20$ (where $m \in \mathbb{R}$) intersect at points A and B, and the tangents to the circles at point A are perpendicular to each other, then the length of segment AB is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_345><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If circle $O_1: x^2+y^2=5$ and circle $O_2: (x-m)^2+y^2=20$ (where $m \in \mathbb{R}$) intersect at points A and B, and the tangents to the circles at point A are perpendicular to each other, then the length of segment AB is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_364><Z_295><Z_158><Z_16><Z_476><Z_2><Z_26><Z_382><Z_160><Z_46><Z_424><ANSWER>

Question: If circle $O_1: x^2+y^2=5$ and circle $O_2: (x-m)^2+y^2=20$ (where $m \in \mathbb{R}$) intersect at points A and B, and the tangents to the circles at point A are perpendicular to each other, then the length of segment AB is \_\_\_\_\_\_.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting with the triple  $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$ , define a sequence of triples  $(x_{n},y_{n},z_{n})$  by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$    $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$   $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for  $n\geq 0$ .Show that each of the sequences  $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$   converges to a limit and find these limits.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting with the triple  $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$ , define a sequence of triples  $(x_{n},y_{n},z_{n})$  by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$    $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$   $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for  $n\geq 0$ .Show that each of the sequences  $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$   converges to a limit and find these limits.<|im_end|>
<|im_start|>assistant
<Z_66><Z_307><Z_415><Z_456><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting with the triple  $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$ , define a sequence of triples  $(x_{n},y_{n},z_{n})$  by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$    $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$   $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for  $n\geq 0$ .Show that each of the sequences  $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$   converges to a limit and find these limits.<|im_end|>
<|im_start|>assistant
<Z_371><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting with the triple  $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$ , define a sequence of triples  $(x_{n},y_{n},z_{n})$  by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$    $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$   $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for  $n\geq 0$ .Show that each of the sequences  $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$   converges to a limit and find these limits.<|im_end|>
<|im_start|>assistant
<Z_75><Z_473><Z_495><Z_444><ANSWER>

Question: Starting with the triple  $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$ , define a sequence of triples  $(x_{n},y_{n},z_{n})$  by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$    $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$   $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for  $n\geq 0$ .Show that each of the sequences  $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$   converges to a limit and find these limits.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a tank in the shape of a cuboid which is 30 centimeters (cm) wide, 20 centimeters (cm) long, and 5 centimeters (cm) high. When pouring water into the tank with a cuboid-shaped bowl measuring 6 centimeters (cm) wide, 4 centimeters (cm) long, and 1 cm (cm high), find out how many times it takes to fill the tank.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a tank in the shape of a cuboid which is 30 centimeters (cm) wide, 20 centimeters (cm) long, and 5 centimeters (cm) high. When pouring water into the tank with a cuboid-shaped bowl measuring 6 centimeters (cm) wide, 4 centimeters (cm) long, and 1 cm (cm high), find out how many times it takes to fill the tank.<|im_end|>
<|im_start|>assistant
<Z_195><Z_143><Z_123><Z_456><Z_106><Z_140><Z_293><Z_434><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a tank in the shape of a cuboid which is 30 centimeters (cm) wide, 20 centimeters (cm) long, and 5 centimeters (cm) high. When pouring water into the tank with a cuboid-shaped bowl measuring 6 centimeters (cm) wide, 4 centimeters (cm) long, and 1 cm (cm high), find out how many times it takes to fill the tank.<|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
There is a tank in the shape of a cuboid which is 30 centimeters (cm) wide, 20 centimeters (cm) long, and 5 centimeters (cm) high. When pouring water into the tank with a cuboid-shaped bowl measuring 6 centimeters (cm) wide, 4 centimeters (cm) long, and 1 cm (cm high), find out how many times it takes to fill the tank.<|im_end|>
<|im_start|>assistant
<Z_20><Z_54><Z_461><Z_395><Z_446><Z_31><Z_181><Z_92><ANSWER>

Question: There is a tank in the shape of a cuboid which is 30 centimeters (cm) wide, 20 centimeters (cm) long, and 5 centimeters (cm) high. When pouring water into the tank with a cuboid-shaped bowl measuring 6 centimeters (cm) wide, 4 centimeters (cm) long, and 1 cm (cm high), find out how many times it takes to fill the tank.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of

$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of

$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$<|im_end|>
<|im_start|>assistant
<Z_66><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of

$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$<|im_end|>
<|im_start|>assistant
<Z_307><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of

$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$<|im_end|>
<|im_start|>assistant
<Z_420><Z_450><ANSWER>

Question:
Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of

$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A right triangle is inscribed in the ellipse whose equation is $9x^2 + y^2 = 36$. The vertex at the right angle of the triangle is at $(0, -6)$, and one of its sides lies along the x-axis. Calculate the square of the hypotenuse length of this triangle, expressing your answer as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.<|im_end|>
<|im_start|>assistant
<Z_176><Z_316><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A right triangle is inscribed in the ellipse whose equation is $9x^2 + y^2 = 36$. The vertex at the right angle of the triangle is at $(0, -6)$, and one of its sides lies along the x-axis. Calculate the square of the hypotenuse length of this triangle, expressing your answer as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.<|im_end|>
<|im_start|>assistant
<Z_66><Z_75><Z_113><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A right triangle is inscribed in the ellipse whose equation is $9x^2 + y^2 = 36$. The vertex at the right angle of the triangle is at $(0, -6)$, and one of its sides lies along the x-axis. Calculate the square of the hypotenuse length of this triangle, expressing your answer as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.<|im_end|>
<|im_start|>assistant
<Z_45><Z_169><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A right triangle is inscribed in the ellipse whose equation is $9x^2 + y^2 = 36$. The vertex at the right angle of the triangle is at $(0, -6)$, and one of its sides lies along the x-axis. Calculate the square of the hypotenuse length of this triangle, expressing your answer as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.<|im_end|>
<|im_start|>assistant
<Z_307><Z_6><Z_195><Z_305><ANSWER>

Question: A right triangle is inscribed in the ellipse whose equation is $9x^2 + y^2 = 36$. The vertex at the right angle of the triangle is at $(0, -6)$, and one of its sides lies along the x-axis. Calculate the square of the hypotenuse length of this triangle, expressing your answer as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Malibu Country Club needs to drain its pool for refinishing. The hose they use to drain it can remove 60 cubic feet of water per minute. The pool is a certain width by 150 feet long by 10 feet deep and is currently at 80% capacity. It takes 1000 minutes to drain the pool. How wide is the pool?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Malibu Country Club needs to drain its pool for refinishing. The hose they use to drain it can remove 60 cubic feet of water per minute. The pool is a certain width by 150 feet long by 10 feet deep and is currently at 80% capacity. It takes 1000 minutes to drain the pool. How wide is the pool?<|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 Malibu Country Club needs to drain its pool for refinishing. The hose they use to drain it can remove 60 cubic feet of water per minute. The pool is a certain width by 150 feet long by 10 feet deep and is currently at 80% capacity. It takes 1000 minutes to drain the pool. How wide is the pool?<|im_end|>
<|im_start|>assistant
<Z_493><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Malibu Country Club needs to drain its pool for refinishing. The hose they use to drain it can remove 60 cubic feet of water per minute. The pool is a certain width by 150 feet long by 10 feet deep and is currently at 80% capacity. It takes 1000 minutes to drain the pool. How wide is the pool?<|im_end|>
<|im_start|>assistant
<Z_1><ANSWER>

Question: The Malibu Country Club needs to drain its pool for refinishing. The hose they use to drain it can remove 60 cubic feet of water per minute. The pool is a certain width by 150 feet long by 10 feet deep and is currently at 80% capacity. It takes 1000 minutes to drain the pool. How wide is the pool?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many numbers are in the list $156, 150, 144, \ldots, 36$?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many numbers are in the list $156, 150, 144, \ldots, 36$?<|im_end|>
<|im_start|>assistant
<Z_66><Z_387><Z_490><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many numbers are in the list $156, 150, 144, \ldots, 36$?<|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
How many numbers are in the list $156, 150, 144, \ldots, 36$?<|im_end|>
<|im_start|>assistant
<Z_23><Z_347><Z_56><ANSWER>

Question: How many numbers are in the list $156, 150, 144, \ldots, 36$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the divisor if the dividend is 167, the quotient is 9, and the remainder is 5?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the divisor if the dividend is 167, the quotient is 9, and the remainder is 5?<|im_end|>
<|im_start|>assistant
<Z_72><Z_446><Z_74><Z_375><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the divisor if the dividend is 167, the quotient is 9, and the remainder is 5?<|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
What is the divisor if the dividend is 167, the quotient is 9, and the remainder is 5?<|im_end|>
<|im_start|>assistant
<Z_334><Z_507><Z_310><Z_273><ANSWER>

Question: What is the divisor if the dividend is 167, the quotient is 9, and the remainder is 5?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janice bought 50 items each priced at 50 cents, 4 dollars, or 5 dollars. If her total purchase price was $\$$100.00, how many 50-cent items did she purchase?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janice bought 50 items each priced at 50 cents, 4 dollars, or 5 dollars. If her total purchase price was $\$$100.00, how many 50-cent items did she purchase?<|im_end|>
<|im_start|>assistant
<Z_171><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janice bought 50 items each priced at 50 cents, 4 dollars, or 5 dollars. If her total purchase price was $\$$100.00, how many 50-cent items did she purchase?<|im_end|>
<|im_start|>assistant
<Z_395><ANSWER>

Question: Janice bought 50 items each priced at 50 cents, 4 dollars, or 5 dollars. If her total purchase price was $\$$100.00, how many 50-cent items did she purchase?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p is the product of the integers from 1 to 34, inclusive, what is the greatest integer k for which 3^k is a factor of p?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p is the product of the integers from 1 to 34, inclusive, what is the greatest integer k for which 3^k is a factor of p?<|im_end|>
<|im_start|>assistant
<Z_176><Z_445><Z_506><Z_365><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p is the product of the integers from 1 to 34, inclusive, what is the greatest integer k for which 3^k is a factor of p?<|im_end|>
<|im_start|>assistant
<Z_127><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p is the product of the integers from 1 to 34, inclusive, what is the greatest integer k for which 3^k is a factor of p?<|im_end|>
<|im_start|>assistant
<Z_430><Z_455><Z_451><Z_344><ANSWER>

Question: If p is the product of the integers from 1 to 34, inclusive, what is the greatest integer k for which 3^k is a factor of p?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A teacher finds that when she offers cookies to her class of 25 students, the mean number of cookies taken by each student is 4. If every student takes at least one cookie, what is the greatest number of cookies one student could have taken?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A teacher finds that when she offers cookies to her class of 25 students, the mean number of cookies taken by each student is 4. If every student takes at least one cookie, what is the greatest number of cookies one student could have taken?<|im_end|>
<|im_start|>assistant
<Z_66><Z_383><Z_341><Z_289><Z_194><Z_180><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A teacher finds that when she offers cookies to her class of 25 students, the mean number of cookies taken by each student is 4. If every student takes at least one cookie, what is the greatest number of cookies one student could have taken?<|im_end|>
<|im_start|>assistant
<Z_160><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A teacher finds that when she offers cookies to her class of 25 students, the mean number of cookies taken by each student is 4. If every student takes at least one cookie, what is the greatest number of cookies one student could have taken?<|im_end|>
<|im_start|>assistant
<Z_188><Z_191><Z_169><Z_265><Z_251><Z_126><Z_452><ANSWER>

Question: A teacher finds that when she offers cookies to her class of 25 students, the mean number of cookies taken by each student is 4. If every student takes at least one cookie, what is the greatest number of cookies one student could have taken?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A, R, M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_2 (AL) + \log_2 (AM) &= 5, \\
\log_2 (ML) + \log_2 (MR) &= 6, \\
\log_2 (RA) + \log_2 (RL) &= 7.
\end{align*}
Compute the value of the product $ARML$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A, R, M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_2 (AL) + \log_2 (AM) &= 5, \\
\log_2 (ML) + \log_2 (MR) &= 6, \\
\log_2 (RA) + \log_2 (RL) &= 7.
\end{align*}
Compute the value of the product $ARML$.<|im_end|>
<|im_start|>assistant
<Z_358><Z_503><Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A, R, M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_2 (AL) + \log_2 (AM) &= 5, \\
\log_2 (ML) + \log_2 (MR) &= 6, \\
\log_2 (RA) + \log_2 (RL) &= 7.
\end{align*}
Compute the value of the product $ARML$.<|im_end|>
<|im_start|>assistant
<Z_174><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A, R, M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_2 (AL) + \log_2 (AM) &= 5, \\
\log_2 (ML) + \log_2 (MR) &= 6, \\
\log_2 (RA) + \log_2 (RL) &= 7.
\end{align*}
Compute the value of the product $ARML$.<|im_end|>
<|im_start|>assistant
<Z_419><Z_307><Z_410><ANSWER>

Question:
Let $A, R, M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_2 (AL) + \log_2 (AM) &= 5, \\
\log_2 (ML) + \log_2 (MR) &= 6, \\
\log_2 (RA) + \log_2 (RL) &= 7.
\end{align*}
Compute the value of the product $ARML$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A set of 8 fixed points is given. For any arbitrarily chosen directed line $\iota$ on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ onto this line be $P_{1}, P_{2}, \cdots, P_{8}$. If these 8 projections do not coincide with each other and are ordered along the direction of line $\iota$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of the integers $1, 2, \cdots, 8$. Define $N_{8}$ as the total number of distinct permutations obtained from the projections of these 8 points onto all directed lines on the plane, denoted as $N(A_{1}, A_{2}, \cdots, A_{8})$. Find the maximum value of $N_{8}$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A set of 8 fixed points is given. For any arbitrarily chosen directed line $\iota$ on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ onto this line be $P_{1}, P_{2}, \cdots, P_{8}$. If these 8 projections do not coincide with each other and are ordered along the direction of line $\iota$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of the integers $1, 2, \cdots, 8$. Define $N_{8}$ as the total number of distinct permutations obtained from the projections of these 8 points onto all directed lines on the plane, denoted as $N(A_{1}, A_{2}, \cdots, A_{8})$. Find the maximum value of $N_{8}$.<|im_end|>
<|im_start|>assistant
<Z_72><Z_300><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A set of 8 fixed points is given. For any arbitrarily chosen directed line $\iota$ on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ onto this line be $P_{1}, P_{2}, \cdots, P_{8}$. If these 8 projections do not coincide with each other and are ordered along the direction of line $\iota$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of the integers $1, 2, \cdots, 8$. Define $N_{8}$ as the total number of distinct permutations obtained from the projections of these 8 points onto all directed lines on the plane, denoted as $N(A_{1}, A_{2}, \cdots, A_{8})$. Find the maximum value of $N_{8}$.<|im_end|>
<|im_start|>assistant
<Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A set of 8 fixed points is given. For any arbitrarily chosen directed line $\iota$ on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ onto this line be $P_{1}, P_{2}, \cdots, P_{8}$. If these 8 projections do not coincide with each other and are ordered along the direction of line $\iota$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of the integers $1, 2, \cdots, 8$. Define $N_{8}$ as the total number of distinct permutations obtained from the projections of these 8 points onto all directed lines on the plane, denoted as $N(A_{1}, A_{2}, \cdots, A_{8})$. Find the maximum value of $N_{8}$.<|im_end|>
<|im_start|>assistant
<Z_82><Z_63><ANSWER>

Question: A set of 8 fixed points is given. For any arbitrarily chosen directed line $\iota$ on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ onto this line be $P_{1}, P_{2}, \cdots, P_{8}$. If these 8 projections do not coincide with each other and are ordered along the direction of line $\iota$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of the integers $1, 2, \cdots, 8$. Define $N_{8}$ as the total number of distinct permutations obtained from the projections of these 8 points onto all directed lines on the plane, denoted as $N(A_{1}, A_{2}, \cdots, A_{8})$. Find the maximum value of $N_{8}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 7 : 12. There are 75 dogs in stock. How many dogs and bunnies are there in total?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 7 : 12. There are 75 dogs in stock. How many dogs and bunnies are there in total?<|im_end|>
<|im_start|>assistant
<Z_195><Z_322><Z_126><Z_130><Z_473><Z_497><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 7 : 12. There are 75 dogs in stock. How many dogs and bunnies are there in total?<|im_end|>
<|im_start|>assistant
<Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 7 : 12. There are 75 dogs in stock. How many dogs and bunnies are there in total?<|im_end|>
<|im_start|>assistant
<Z_124><Z_4><Z_2><Z_479><Z_108><Z_479><Z_392><ANSWER>

Question: In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 7 : 12. There are 75 dogs in stock. How many dogs and bunnies are there in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rhombus has an area of some value, one diagonal is 36 cm, and the other diagonal is 2400 cm. What is the area of the rhombus?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rhombus has an area of some value, one diagonal is 36 cm, and the other diagonal is 2400 cm. What is the area of the rhombus?<|im_end|>
<|im_start|>assistant
<Z_195><Z_79><Z_205><Z_339><Z_53><Z_192><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rhombus has an area of some value, one diagonal is 36 cm, and the other diagonal is 2400 cm. What is the area of the rhombus?<|im_end|>
<|im_start|>assistant
<Z_262><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rhombus has an area of some value, one diagonal is 36 cm, and the other diagonal is 2400 cm. What is the area of the rhombus?<|im_end|>
<|im_start|>assistant
<Z_22><Z_228><Z_95><Z_437><Z_119><Z_150><ANSWER>

Question: A rhombus has an area of some value, one diagonal is 36 cm, and the other diagonal is 2400 cm. What is the area of the rhombus?


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

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school with a total of 1600 male and female students conducted a health survey, selecting a sample of 200 using a stratified sampling method. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ______.<|im_end|>
<|im_start|>assistant
<Z_358><Z_182><Z_267><Z_171><Z_291><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school with a total of 1600 male and female students conducted a health survey, selecting a sample of 200 using a stratified sampling method. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ______.<|im_end|>
<|im_start|>assistant
<Z_196><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school with a total of 1600 male and female students conducted a health survey, selecting a sample of 200 using a stratified sampling method. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ______.<|im_end|>
<|im_start|>assistant
<Z_505><Z_382><Z_309><Z_179><Z_282><ANSWER>

Question: A school with a total of 1600 male and female students conducted a health survey, selecting a sample of 200 using a stratified sampling method. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a : b = 3 : 4 , b : c = 7 : 9 , c : d = 5 : 7 , what is the ratio of a to d?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a : b = 3 : 4 , b : c = 7 : 9 , c : d = 5 : 7 , what is the ratio of a to d?<|im_end|>
<|im_start|>assistant
<Z_66><Z_215><Z_235><Z_158><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a : b = 3 : 4 , b : c = 7 : 9 , c : d = 5 : 7 , what is the ratio of a to d?<|im_end|>
<|im_start|>assistant
<Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a : b = 3 : 4 , b : c = 7 : 9 , c : d = 5 : 7 , what is the ratio of a to d?<|im_end|>
<|im_start|>assistant
<Z_189><Z_226><Z_170><Z_55><ANSWER>

Question: If a : b = 3 : 4 , b : c = 7 : 9 , c : d = 5 : 7 , what is the ratio of a to d?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 1 - \sin \left(\frac{\pi}{3}x + \frac{\pi}{6}\right)$, find $f(1) + f(2) + f(3) + \ldots + f(2023)$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 1 - \sin \left(\frac{\pi}{3}x + \frac{\pi}{6}\right)$, find $f(1) + f(2) + f(3) + \ldots + f(2023)$.<|im_end|>
<|im_start|>assistant
<Z_66><Z_125><Z_337><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 1 - \sin \left(\frac{\pi}{3}x + \frac{\pi}{6}\right)$, find $f(1) + f(2) + f(3) + \ldots + f(2023)$.<|im_end|>
<|im_start|>assistant
<Z_147><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 1 - \sin \left(\frac{\pi}{3}x + \frac{\pi}{6}\right)$, find $f(1) + f(2) + f(3) + \ldots + f(2023)$.<|im_end|>
<|im_start|>assistant
<Z_293><Z_81><Z_473><ANSWER>

Question: Given the function $f(x) = 1 - \sin \left(\frac{\pi}{3}x + \frac{\pi}{6}\right)$, find $f(1) + f(2) + f(3) + \ldots + f(2023)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a and b can finish a work in 12 days while a alone can do the same work in 24 days . in how many days b alone will complete the work ?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a and b can finish a work in 12 days while a alone can do the same work in 24 days . in how many days b alone will complete the work ?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a and b can finish a work in 12 days while a alone can do the same work in 24 days . in how many days b alone will complete the work ?<|im_end|>
<|im_start|>assistant
<Z_303><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a and b can finish a work in 12 days while a alone can do the same work in 24 days . in how many days b alone will complete the work ?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

Question: a and b can finish a work in 12 days while a alone can do the same work in 24 days . in how many days b alone will complete the work ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the figure, two pairs of identical isosceles right triangles are cut off of square $WXYZ$, leaving rectangle $MNPQ$. The total area removed is $180 \text{ m}^2$. What is the length of $MQ$, in meters?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the figure, two pairs of identical isosceles right triangles are cut off of square $WXYZ$, leaving rectangle $MNPQ$. The total area removed is $180 \text{ m}^2$. What is the length of $MQ$, in meters?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the figure, two pairs of identical isosceles right triangles are cut off of square $WXYZ$, leaving rectangle $MNPQ$. The total area removed is $180 \text{ m}^2$. What is the length of $MQ$, in meters?<|im_end|>
<|im_start|>assistant
<Z_284><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the figure, two pairs of identical isosceles right triangles are cut off of square $WXYZ$, leaving rectangle $MNPQ$. The total area removed is $180 \text{ m}^2$. What is the length of $MQ$, in meters?<|im_end|>
<|im_start|>assistant
<Z_103><ANSWER>

Question: In the figure, two pairs of identical isosceles right triangles are cut off of square $WXYZ$, leaving rectangle $MNPQ$. The total area removed is $180 \text{ m}^2$. What is the length of $MQ$, in meters?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$?<|im_end|>
<|im_start|>assistant
<Z_195><Z_38><Z_316><Z_310><Z_452><Z_354><Z_193><Z_80><Z_417><Z_299><Z_322><Z_299><Z_400><Z_477><Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$?<|im_end|>
<|im_start|>assistant
<Z_300><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$?<|im_end|>
<|im_start|>assistant
<Z_251><Z_169><Z_445><Z_292><Z_326><Z_92><Z_217><Z_90><Z_96><Z_435><Z_200><Z_403><Z_467><Z_369><Z_455><ANSWER>

Question: What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains start from the same station, one is travelling at a certain speed and the other one at 31 mph along parallel tracks in the same direction. The distance between the two trains after 8 hours is 160 miles. What is the speed of the first train?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains start from the same station, one is travelling at a certain speed and the other one at 31 mph along parallel tracks in the same direction. The distance between the two trains after 8 hours is 160 miles. What is the speed of the first train?<|im_end|>
<|im_start|>assistant
<Z_195><Z_502><Z_488><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains start from the same station, one is travelling at a certain speed and the other one at 31 mph along parallel tracks in the same direction. The distance between the two trains after 8 hours is 160 miles. What is the speed of the first train?<|im_end|>
<|im_start|>assistant
<Z_8><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains start from the same station, one is travelling at a certain speed and the other one at 31 mph along parallel tracks in the same direction. The distance between the two trains after 8 hours is 160 miles. What is the speed of the first train?<|im_end|>
<|im_start|>assistant
<Z_155><Z_84><Z_15><Z_101><ANSWER>

Question: Two trains start from the same station, one is travelling at a certain speed and the other one at 31 mph along parallel tracks in the same direction. The distance between the two trains after 8 hours is 160 miles. What is the speed of the first train?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest integer less than 61 that leaves a remainder of 5 when divided by 6.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest integer less than 61 that leaves a remainder of 5 when divided by 6.<|im_end|>
<|im_start|>assistant
<Z_5><Z_465><Z_295><Z_503><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 less than 61 that leaves a remainder of 5 when divided by 6.<|im_end|>
<|im_start|>assistant
<Z_262><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 less than 61 that leaves a remainder of 5 when divided by 6.<|im_end|>
<|im_start|>assistant
<Z_433><Z_270><Z_405><Z_280><ANSWER>

Question: Find the largest integer less than 61 that leaves a remainder of 5 when divided by 6.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A library has an average of 510 visitors on Sundays and some visitors on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is 285. How many visitors does the library have on average on other days?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A library has an average of 510 visitors on Sundays and some visitors on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is 285. How many visitors does the library have on average on other days?<|im_end|>
<|im_start|>assistant
<Z_358><Z_473><Z_268><Z_472><Z_59><Z_497><Z_226><Z_273><Z_255><Z_241><Z_38><Z_452><Z_412><Z_255><Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A library has an average of 510 visitors on Sundays and some visitors on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is 285. How many visitors does the library have on average on other days?<|im_end|>
<|im_start|>assistant
<Z_308><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A library has an average of 510 visitors on Sundays and some visitors on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is 285. How many visitors does the library have on average on other days?<|im_end|>
<|im_start|>assistant
<Z_189><Z_164><Z_431><Z_109><Z_65><Z_444><Z_504><Z_490><Z_15><Z_172><Z_209><Z_492><Z_73><Z_200><Z_510><ANSWER>

Question: A library has an average of 510 visitors on Sundays and some visitors on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is 285. How many visitors does the library have on average on other days?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 pupils in total. Out of them, some pupils like pizza, 115 like burgers, and 40 pupils like both pizza and burgers. How many pupils like pizza?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 pupils in total. Out of them, some pupils like pizza, 115 like burgers, and 40 pupils like both pizza and burgers. How many pupils like pizza?<|im_end|>
<|im_start|>assistant
<Z_72><Z_334><Z_220><Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 pupils in total. Out of them, some pupils like pizza, 115 like burgers, and 40 pupils like both pizza and burgers. How many pupils like pizza?<|im_end|>
<|im_start|>assistant
<Z_491><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 pupils in total. Out of them, some pupils like pizza, 115 like burgers, and 40 pupils like both pizza and burgers. How many pupils like pizza?<|im_end|>
<|im_start|>assistant
<Z_186><Z_266><Z_240><Z_10><ANSWER>

Question: There are 200 pupils in total. Out of them, some pupils like pizza, 115 like burgers, and 40 pupils like both pizza and burgers. How many pupils like pizza?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The speed of a car is 90 km in the first hour and some km in the second hour. The average speed of the car is 82.5 km/h. What is the speed of the car in the second hour?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The speed of a car is 90 km in the first hour and some km in the second hour. The average speed of the car is 82.5 km/h. What is the speed of the car in the second hour?<|im_end|>
<|im_start|>assistant
<Z_195><Z_177><Z_463><Z_501><Z_275><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The speed of a car is 90 km in the first hour and some km in the second hour. The average speed of the car is 82.5 km/h. What is the speed of the car in the second hour?<|im_end|>
<|im_start|>assistant
<Z_349><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The speed of a car is 90 km in the first hour and some km in the second hour. The average speed of the car is 82.5 km/h. What is the speed of the car in the second hour?<|im_end|>
<|im_start|>assistant
<Z_326><Z_498><Z_84><Z_449><Z_83><ANSWER>

Question: The speed of a car is 90 km in the first hour and some km in the second hour. The average speed of the car is 82.5 km/h. What is the speed of the car in the second hour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$<|im_end|>
<|im_start|>assistant
<Z_66><Z_190><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$<|im_end|>
<|im_start|>assistant
<Z_221><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$<|im_end|>
<|im_start|>assistant
<Z_220><Z_326><Z_24><ANSWER>

Question: Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Austin receives $5 for every hour he helps his father build a deck in their backyard. He works for 2 hours on Mondays, an hour on Wednesdays, and a certain number of hours on Fridays. If Austin wants to buy a bicycle that costs $180, and he has to work for 6 weeks, how many hours does he work on Fridays?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Austin receives $5 for every hour he helps his father build a deck in their backyard. He works for 2 hours on Mondays, an hour on Wednesdays, and a certain number of hours on Fridays. If Austin wants to buy a bicycle that costs $180, and he has to work for 6 weeks, how many hours does he work on Fridays?<|im_end|>
<|im_start|>assistant
<Z_176><Z_509><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Austin receives $5 for every hour he helps his father build a deck in their backyard. He works for 2 hours on Mondays, an hour on Wednesdays, and a certain number of hours on Fridays. If Austin wants to buy a bicycle that costs $180, and he has to work for 6 weeks, how many hours does he work on Fridays?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Austin receives $5 for every hour he helps his father build a deck in their backyard. He works for 2 hours on Mondays, an hour on Wednesdays, and a certain number of hours on Fridays. If Austin wants to buy a bicycle that costs $180, and he has to work for 6 weeks, how many hours does he work on Fridays?<|im_end|>
<|im_start|>assistant
<Z_206><Z_278><ANSWER>

Question: Austin receives $5 for every hour he helps his father build a deck in their backyard. He works for 2 hours on Mondays, an hour on Wednesdays, and a certain number of hours on Fridays. If Austin wants to buy a bicycle that costs $180, and he has to work for 6 weeks, how many hours does he work on Fridays?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of investments of two partners p and q is 7 : 5 and the ratio of their profits is 7 : 14 . if p invested the money for 5 months , find for how much time did q invest the money ?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of investments of two partners p and q is 7 : 5 and the ratio of their profits is 7 : 14 . if p invested the money for 5 months , find for how much time did q invest the money ?<|im_end|>
<|im_start|>assistant
<Z_66><Z_405><Z_213><Z_231><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of investments of two partners p and q is 7 : 5 and the ratio of their profits is 7 : 14 . if p invested the money for 5 months , find for how much time did q invest the money ?<|im_end|>
<|im_start|>assistant
<Z_481><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of investments of two partners p and q is 7 : 5 and the ratio of their profits is 7 : 14 . if p invested the money for 5 months , find for how much time did q invest the money ?<|im_end|>
<|im_start|>assistant
<Z_244><Z_94><Z_320><Z_28><Z_353><ANSWER>

Question: the ratio of investments of two partners p and q is 7 : 5 and the ratio of their profits is 7 : 14 . if p invested the money for 5 months , find for how much time did q invest the money ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes John 45 minutes to cycle uphill 2 km from his home to the local library, but it only takes him 15 minutes to cycle back home along the same route. What is his average speed, in km/hr, for the entire journey?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes John 45 minutes to cycle uphill 2 km from his home to the local library, but it only takes him 15 minutes to cycle back home along the same route. What is his average speed, in km/hr, for the entire journey?<|im_end|>
<|im_start|>assistant
<Z_5><Z_446><Z_268><Z_376><Z_99><Z_330><Z_456><Z_106><Z_133><Z_265><Z_425><Z_400><Z_98><Z_215><Z_269><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes John 45 minutes to cycle uphill 2 km from his home to the local library, but it only takes him 15 minutes to cycle back home along the same route. What is his average speed, in km/hr, for the entire journey?<|im_end|>
<|im_start|>assistant
<Z_403><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes John 45 minutes to cycle uphill 2 km from his home to the local library, but it only takes him 15 minutes to cycle back home along the same route. What is his average speed, in km/hr, for the entire journey?<|im_end|>
<|im_start|>assistant
<Z_108><Z_149><Z_33><Z_229><Z_53><Z_501><Z_353><Z_473><Z_369><Z_357><Z_358><Z_329><Z_424><Z_43><Z_199><ANSWER>

Question: It takes John 45 minutes to cycle uphill 2 km from his home to the local library, but it only takes him 15 minutes to cycle back home along the same route. What is his average speed, in km/hr, for the entire journey?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 2, 3, 4, 5, and 6?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 2, 3, 4, 5, and 6?<|im_end|>
<|im_start|>assistant
<Z_72><Z_475><Z_330><Z_305><Z_41><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 2, 3, 4, 5, and 6?<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 2, 3, 4, 5, and 6?<|im_end|>
<|im_start|>assistant
<Z_275><Z_104><Z_168><Z_334><Z_362><Z_389><ANSWER>

Question: What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 2, 3, 4, 5, and 6?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a checkered square $7 \times 7$, an equal number of $2 \x 2$ squares and $1 \times 4$ rectangles were cut along the cell boundaries. What is the maximum number of such figures that could have been cut out?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a checkered square $7 \times 7$, an equal number of $2 \x 2$ squares and $1 \times 4$ rectangles were cut along the cell boundaries. What is the maximum number of such figures that could have been cut out?<|im_end|>
<|im_start|>assistant
<Z_176><Z_356><Z_219><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a checkered square $7 \times 7$, an equal number of $2 \x 2$ squares and $1 \times 4$ rectangles were cut along the cell boundaries. What is the maximum number of such figures that could have been cut out?<|im_end|>
<|im_start|>assistant
<Z_241><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a checkered square $7 \times 7$, an equal number of $2 \x 2$ squares and $1 \times 4$ rectangles were cut along the cell boundaries. What is the maximum number of such figures that could have been cut out?<|im_end|>
<|im_start|>assistant
<Z_428><Z_450><Z_492><ANSWER>

Question: From a checkered square $7 \times 7$, an equal number of $2 \x 2$ squares and $1 \times 4$ rectangles were cut along the cell boundaries. What is the maximum number of such figures that could have been cut out?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Assume that $x_1, x_2, \ldots, x_7$ are real numbers such that
\[
\begin{aligned}
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 + 11x_6 + 13x_7 &= 3, \\
3x_1 + 5x_2 + 7x_3 + 9x_4 + 11x_5 + 13x_6 + 15x_7 &= 15, \\
5x_1 + 7x_2 + 9x_3 + 11x_4 + 13x_5 + 15x_6 + 17x_7 &= 85.
\end{aligned}
\]
Find the value of $7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Assume that $x_1, x_2, \ldots, x_7$ are real numbers such that
\[
\begin{aligned}
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 + 11x_6 + 13x_7 &= 3, \\
3x_1 + 5x_2 + 7x_3 + 9x_4 + 11x_5 + 13x_6 + 15x_7 &= 15, \\
5x_1 + 7x_2 + 9x_3 + 11x_4 + 13x_5 + 15x_6 + 17x_7 &= 85.
\end{aligned}
\]
Find the value of $7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7$.<|im_end|>
<|im_start|>assistant
<Z_358><Z_160><Z_80><Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Assume that $x_1, x_2, \ldots, x_7$ are real numbers such that
\[
\begin{aligned}
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 + 11x_6 + 13x_7 &= 3, \\
3x_1 + 5x_2 + 7x_3 + 9x_4 + 11x_5 + 13x_6 + 15x_7 &= 15, \\
5x_1 + 7x_2 + 9x_3 + 11x_4 + 13x_5 + 15x_6 + 17x_7 &= 85.
\end{aligned}
\]
Find the value of $7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7$.<|im_end|>
<|im_start|>assistant
<Z_33><ANSWER>

Question:
Assume that $x_1, x_2, \ldots, x_7$ are real numbers such that
\[
\begin{aligned}
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 + 11x_6 + 13x_7 &= 3, \\
3x_1 + 5x_2 + 7x_3 + 9x_4 + 11x_5 + 13x_6 + 15x_7 &= 15, \\
5x_1 + 7x_2 + 9x_3 + 11x_4 + 13x_5 + 15x_6 + 17x_7 &= 85.
\end{aligned}
\]
Find the value of $7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that has exactly ten distinct positive factors?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that has exactly ten distinct positive factors?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that has exactly ten distinct positive factors?<|im_end|>
<|im_start|>assistant
<Z_338><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that has exactly ten distinct positive factors?<|im_end|>
<|im_start|>assistant
<Z_383><ANSWER>

Question: What is the smallest positive integer that has exactly ten distinct positive factors?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the base-3 expression for $345_{10}$?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the base-3 expression for $345_{10}$?<|im_end|>
<|im_start|>assistant
<Z_66><Z_446><Z_141><Z_458><Z_490><Z_295><Z_267><Z_497><Z_160><Z_289><Z_294><Z_199><Z_418><Z_462><Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the base-3 expression for $345_{10}$?<|im_end|>
<|im_start|>assistant
<Z_363><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the base-3 expression for $345_{10}$?<|im_end|>
<|im_start|>assistant
<Z_117><Z_218><Z_325><Z_0><Z_44><Z_106><Z_288><Z_375><Z_127><Z_115><Z_52><Z_476><Z_62><Z_167><Z_188><ANSWER>

Question: What is the sum of the digits of the base-3 expression for $345_{10}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of 3 numbers is 5 : 3 : 4 and their sum is 108. What is the second number of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of 3 numbers is 5 : 3 : 4 and their sum is 108. What is the second number of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_176><Z_337><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of 3 numbers is 5 : 3 : 4 and their sum is 108. What is the second number of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_260><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of 3 numbers is 5 : 3 : 4 and their sum is 108. What is the second number of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_262><Z_375><ANSWER>

Question: The ratio of 3 numbers is 5 : 3 : 4 and their sum is 108. What is the second number of the 3 numbers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bruce purchased 8 kg of grapes at the rate of 70 per kg and 8 kg of mangoes at the rate of 55 per kg. How much amount did he pay to the shopkeeper?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bruce purchased 8 kg of grapes at the rate of 70 per kg and 8 kg of mangoes at the rate of 55 per kg. How much amount did he pay to the shopkeeper?<|im_end|>
<|im_start|>assistant
<Z_176><Z_42><Z_165><Z_364><Z_389><Z_503><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bruce purchased 8 kg of grapes at the rate of 70 per kg and 8 kg of mangoes at the rate of 55 per kg. How much amount did he pay to the shopkeeper?<|im_end|>
<|im_start|>assistant
<Z_249><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bruce purchased 8 kg of grapes at the rate of 70 per kg and 8 kg of mangoes at the rate of 55 per kg. How much amount did he pay to the shopkeeper?<|im_end|>
<|im_start|>assistant
<Z_92><Z_128><Z_396><Z_442><Z_454><Z_343><ANSWER>

Question: Bruce purchased 8 kg of grapes at the rate of 70 per kg and 8 kg of mangoes at the rate of 55 per kg. How much amount did he pay to the shopkeeper?