<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|im_end|>
<|im_start|>assistant
<Z_477><Z_345><Z_205><Z_294><Z_322><Z_436><Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|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
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|im_end|>
<|im_start|>assistant
<Z_176><Z_425><Z_305><Z_137><Z_388><Z_125><Z_220><ANSWER>

Question: Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_285><Z_341><Z_348><Z_475><Z_502><Z_439><Z_335><Z_211><Z_115><Z_125><Z_218><Z_393><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_29><Z_508><Z_438><Z_165><Z_418><Z_272><Z_326><Z_479><Z_284><Z_7><Z_48><Z_262><Z_184><ANSWER>

Question: Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_369><Z_264><Z_232><Z_92><Z_223><Z_499><Z_332><Z_268><Z_469><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_406><Z_427><Z_91><Z_357><Z_505><Z_390><Z_172><Z_306><Z_332><ANSWER>

Question: What is the greatest possible three-digit number that is divisible by 3, 6, and 4?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_477><Z_250><Z_373><Z_391><Z_269><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|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
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_167><Z_418><Z_407><Z_1><Z_282><Z_31><ANSWER>

Question:
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_369><Z_485><Z_63><Z_506><Z_170><Z_116><Z_165><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_412><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_2><Z_375><Z_393><Z_362><Z_486><Z_244><Z_490><ANSWER>

Question: Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_477><Z_321><Z_77><Z_67><Z_464><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_412><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_475><Z_470><Z_368><Z_145><Z_67><ANSWER>

Question: How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_191><ANSWER>

Question: A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three baskets A, B, and C contain different numbers of differently colored marbles. Basket A contains 7 red marbles, 3 yellow marbles, 5 blue marbles, and 6 purple marbles. Basket B contains 10 green marbles, 1 yellow marble, 2 orange marbles, and 5 red marbles. Basket C contains 3 white marbles, 9 yellow marbles, 4 black marbles, and 2 green marbles. What is the difference between the total number of red and green marbles in the basket with the smallest difference in the number of blue and orange marbles?<|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
Three baskets A, B, and C contain different numbers of differently colored marbles. Basket A contains 7 red marbles, 3 yellow marbles, 5 blue marbles, and 6 purple marbles. Basket B contains 10 green marbles, 1 yellow marble, 2 orange marbles, and 5 red marbles. Basket C contains 3 white marbles, 9 yellow marbles, 4 black marbles, and 2 green marbles. What is the difference between the total number of red and green marbles in the basket with the smallest difference in the number of blue and orange marbles?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three baskets A, B, and C contain different numbers of differently colored marbles. Basket A contains 7 red marbles, 3 yellow marbles, 5 blue marbles, and 6 purple marbles. Basket B contains 10 green marbles, 1 yellow marble, 2 orange marbles, and 5 red marbles. Basket C contains 3 white marbles, 9 yellow marbles, 4 black marbles, and 2 green marbles. What is the difference between the total number of red and green marbles in the basket with the smallest difference in the number of blue and orange marbles?<|im_end|>
<|im_start|>assistant
<Z_359><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three baskets A, B, and C contain different numbers of differently colored marbles. Basket A contains 7 red marbles, 3 yellow marbles, 5 blue marbles, and 6 purple marbles. Basket B contains 10 green marbles, 1 yellow marble, 2 orange marbles, and 5 red marbles. Basket C contains 3 white marbles, 9 yellow marbles, 4 black marbles, and 2 green marbles. What is the difference between the total number of red and green marbles in the basket with the smallest difference in the number of blue and orange marbles?<|im_end|>
<|im_start|>assistant
<Z_285><ANSWER>

Question: Three baskets A, B, and C contain different numbers of differently colored marbles. Basket A contains 7 red marbles, 3 yellow marbles, 5 blue marbles, and 6 purple marbles. Basket B contains 10 green marbles, 1 yellow marble, 2 orange marbles, and 5 red marbles. Basket C contains 3 white marbles, 9 yellow marbles, 4 black marbles, and 2 green marbles. What is the difference between the total number of red and green marbles in the basket with the smallest difference in the number of blue and orange marbles?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?

*Proposed by Ray Li*<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?

*Proposed by Ray Li*<|im_end|>
<|im_start|>assistant
<Z_24><Z_342><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?

*Proposed by Ray Li*<|im_end|>
<|im_start|>assistant
<Z_504><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?

*Proposed by Ray Li*<|im_end|>
<|im_start|>assistant
<Z_399><Z_463><ANSWER>

Question:
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?

*Proposed by Ray Li*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each term of a sequence, after the first term, is inversely proportional to the term preceding it with a constant of proportionality that remains constant throughout the sequence. If the first term is 3 and the second term is 6, what is the 11th term of the sequence?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each term of a sequence, after the first term, is inversely proportional to the term preceding it with a constant of proportionality that remains constant throughout the sequence. If the first term is 3 and the second term is 6, what is the 11th term of the sequence?<|im_end|>
<|im_start|>assistant
<Z_369><Z_386><Z_134><Z_421><Z_161><Z_211><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each term of a sequence, after the first term, is inversely proportional to the term preceding it with a constant of proportionality that remains constant throughout the sequence. If the first term is 3 and the second term is 6, what is the 11th term of the sequence?<|im_end|>
<|im_start|>assistant
<Z_490><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each term of a sequence, after the first term, is inversely proportional to the term preceding it with a constant of proportionality that remains constant throughout the sequence. If the first term is 3 and the second term is 6, what is the 11th term of the sequence?<|im_end|>
<|im_start|>assistant
<Z_438><Z_245><Z_96><Z_406><Z_448><Z_21><Z_379><ANSWER>

Question: Each term of a sequence, after the first term, is inversely proportional to the term preceding it with a constant of proportionality that remains constant throughout the sequence. If the first term is 3 and the second term is 6, what is the 11th term of the sequence?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$.  If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$.  If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_320><Z_443><Z_248><Z_220><Z_330><Z_485><Z_53><Z_247><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$.  If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$?<|im_end|>
<|im_start|>assistant
<Z_62><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$.  If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$?<|im_end|>
<|im_start|>assistant
<Z_346><Z_347><Z_5><Z_17><Z_108><Z_200><Z_121><Z_429><Z_94><Z_413><ANSWER>

Question: In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$.  If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Natural numbers of the form \(F_n=2^{2^{2n}} + 1 \) are special forms of modified Fermat numbers. Calculate the units digit of \(F_{500}\).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Natural numbers of the form \(F_n=2^{2^{2n}} + 1 \) are special forms of modified Fermat numbers. Calculate the units digit of \(F_{500}\).<|im_end|>
<|im_start|>assistant
<Z_369><Z_53><Z_333><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Natural numbers of the form \(F_n=2^{2^{2n}} + 1 \) are special forms of modified Fermat numbers. Calculate the units digit of \(F_{500}\).<|im_end|>
<|im_start|>assistant
<Z_91><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Natural numbers of the form \(F_n=2^{2^{2n}} + 1 \) are special forms of modified Fermat numbers. Calculate the units digit of \(F_{500}\).<|im_end|>
<|im_start|>assistant
<Z_452><Z_502><Z_224><ANSWER>

Question: Natural numbers of the form \(F_n=2^{2^{2n}} + 1 \) are special forms of modified Fermat numbers. Calculate the units digit of \(F_{500}\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kaiden is collecting cans of soup for the local soup kitchen. He collects 158 cans during his first week and increases his collection rate by 25% in the second week. If his goal is to collect 500 cans in a total of three weeks, how many cans of soup does he need to collect in the third week?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kaiden is collecting cans of soup for the local soup kitchen. He collects 158 cans during his first week and increases his collection rate by 25% in the second week. If his goal is to collect 500 cans in a total of three weeks, how many cans of soup does he need to collect in the third week?<|im_end|>
<|im_start|>assistant
<Z_477><Z_146><Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kaiden is collecting cans of soup for the local soup kitchen. He collects 158 cans during his first week and increases his collection rate by 25% in the second week. If his goal is to collect 500 cans in a total of three weeks, how many cans of soup does he need to collect in the third week?<|im_end|>
<|im_start|>assistant
<Z_239><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kaiden is collecting cans of soup for the local soup kitchen. He collects 158 cans during his first week and increases his collection rate by 25% in the second week. If his goal is to collect 500 cans in a total of three weeks, how many cans of soup does he need to collect in the third week?<|im_end|>
<|im_start|>assistant
<Z_492><Z_490><Z_123><ANSWER>

Question: Kaiden is collecting cans of soup for the local soup kitchen. He collects 158 cans during his first week and increases his collection rate by 25% in the second week. If his goal is to collect 500 cans in a total of three weeks, how many cans of soup does he need to collect in the third week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mikaela was repainting her irregularly-shaped bathroom. She initially bought 16 containers of paint to cover the bathroom's four walls, which were not equally sized. At the last minute, she decided to put tile on one wall and paint flowers on the ceiling with one container of paint instead. Furthermore, she decides to create a gradient effect on the remaining walls, requiring an additional container of paint for each wall. Taking into consideration the various complexities of the room, how many containers of paint will she have left over?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mikaela was repainting her irregularly-shaped bathroom. She initially bought 16 containers of paint to cover the bathroom's four walls, which were not equally sized. At the last minute, she decided to put tile on one wall and paint flowers on the ceiling with one container of paint instead. Furthermore, she decides to create a gradient effect on the remaining walls, requiring an additional container of paint for each wall. Taking into consideration the various complexities of the room, how many containers of paint will she have left over?<|im_end|>
<|im_start|>assistant
<Z_477><Z_284><Z_322><Z_157><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mikaela was repainting her irregularly-shaped bathroom. She initially bought 16 containers of paint to cover the bathroom's four walls, which were not equally sized. At the last minute, she decided to put tile on one wall and paint flowers on the ceiling with one container of paint instead. Furthermore, she decides to create a gradient effect on the remaining walls, requiring an additional container of paint for each wall. Taking into consideration the various complexities of the room, how many containers of paint will she have left over?<|im_end|>
<|im_start|>assistant
<Z_132><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mikaela was repainting her irregularly-shaped bathroom. She initially bought 16 containers of paint to cover the bathroom's four walls, which were not equally sized. At the last minute, she decided to put tile on one wall and paint flowers on the ceiling with one container of paint instead. Furthermore, she decides to create a gradient effect on the remaining walls, requiring an additional container of paint for each wall. Taking into consideration the various complexities of the room, how many containers of paint will she have left over?<|im_end|>
<|im_start|>assistant
<Z_90><Z_245><Z_391><Z_70><ANSWER>

Question: Mikaela was repainting her irregularly-shaped bathroom. She initially bought 16 containers of paint to cover the bathroom's four walls, which were not equally sized. At the last minute, she decided to put tile on one wall and paint flowers on the ceiling with one container of paint instead. Furthermore, she decides to create a gradient effect on the remaining walls, requiring an additional container of paint for each wall. Taking into consideration the various complexities of the room, how many containers of paint will she have left over?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carlson has three boxes, each containing 10 candies. One box is labeled with the number 4, another with the number 7, and the third with the number 10. In one operation, Carlson performs the following two steps sequentially:

- Takes from any box the number of candies equal to the number written on it.
- Eats 3 candies from those taken, and puts the remaining candies into any other box.

What is the maximum number of candies Carlson can eat as a result of several such operations?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carlson has three boxes, each containing 10 candies. One box is labeled with the number 4, another with the number 7, and the third with the number 10. In one operation, Carlson performs the following two steps sequentially:

- Takes from any box the number of candies equal to the number written on it.
- Eats 3 candies from those taken, and puts the remaining candies into any other box.

What is the maximum number of candies Carlson can eat as a result of several such operations?<|im_end|>
<|im_start|>assistant
<Z_24><Z_298><Z_240><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carlson has three boxes, each containing 10 candies. One box is labeled with the number 4, another with the number 7, and the third with the number 10. In one operation, Carlson performs the following two steps sequentially:

- Takes from any box the number of candies equal to the number written on it.
- Eats 3 candies from those taken, and puts the remaining candies into any other box.

What is the maximum number of candies Carlson can eat as a result of several such operations?<|im_end|>
<|im_start|>assistant
<Z_346><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carlson has three boxes, each containing 10 candies. One box is labeled with the number 4, another with the number 7, and the third with the number 10. In one operation, Carlson performs the following two steps sequentially:

- Takes from any box the number of candies equal to the number written on it.
- Eats 3 candies from those taken, and puts the remaining candies into any other box.

What is the maximum number of candies Carlson can eat as a result of several such operations?<|im_end|>
<|im_start|>assistant
<Z_12><Z_215><Z_86><ANSWER>

Question:
Carlson has three boxes, each containing 10 candies. One box is labeled with the number 4, another with the number 7, and the third with the number 10. In one operation, Carlson performs the following two steps sequentially:

- Takes from any box the number of candies equal to the number written on it.
- Eats 3 candies from those taken, and puts the remaining candies into any other box.

What is the maximum number of candies Carlson can eat as a result of several such operations?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Pogo, the four-legged martian, walks across the soft Martian soil, it leaves some footprints every meter. Grimzi, the three-legged Plutonian, leaves 3 footprints in the soft sands of Pluto for every 6 meters it walks. Pogo travels 6000 meters across the soil of Mars, and Grimzi travels for 6000 meters across the fine sands of Pluto. The combined total number of footprints the two creatures will leave on the surfaces of their respective planets is 27,000. How many footprints does Pogo leave every meter?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Pogo, the four-legged martian, walks across the soft Martian soil, it leaves some footprints every meter. Grimzi, the three-legged Plutonian, leaves 3 footprints in the soft sands of Pluto for every 6 meters it walks. Pogo travels 6000 meters across the soil of Mars, and Grimzi travels for 6000 meters across the fine sands of Pluto. The combined total number of footprints the two creatures will leave on the surfaces of their respective planets is 27,000. How many footprints does Pogo leave every meter?<|im_end|>
<|im_start|>assistant
<Z_477><Z_45><Z_488><Z_199><Z_99><Z_372><Z_322><Z_304><Z_303><Z_425><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Pogo, the four-legged martian, walks across the soft Martian soil, it leaves some footprints every meter. Grimzi, the three-legged Plutonian, leaves 3 footprints in the soft sands of Pluto for every 6 meters it walks. Pogo travels 6000 meters across the soil of Mars, and Grimzi travels for 6000 meters across the fine sands of Pluto. The combined total number of footprints the two creatures will leave on the surfaces of their respective planets is 27,000. How many footprints does Pogo leave every meter?<|im_end|>
<|im_start|>assistant
<Z_133><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Pogo, the four-legged martian, walks across the soft Martian soil, it leaves some footprints every meter. Grimzi, the three-legged Plutonian, leaves 3 footprints in the soft sands of Pluto for every 6 meters it walks. Pogo travels 6000 meters across the soil of Mars, and Grimzi travels for 6000 meters across the fine sands of Pluto. The combined total number of footprints the two creatures will leave on the surfaces of their respective planets is 27,000. How many footprints does Pogo leave every meter?<|im_end|>
<|im_start|>assistant
<Z_456><Z_105><Z_208><Z_391><Z_41><Z_186><Z_88><Z_263><Z_250><Z_373><ANSWER>

Question: When Pogo, the four-legged martian, walks across the soft Martian soil, it leaves some footprints every meter. Grimzi, the three-legged Plutonian, leaves 3 footprints in the soft sands of Pluto for every 6 meters it walks. Pogo travels 6000 meters across the soil of Mars, and Grimzi travels for 6000 meters across the fine sands of Pluto. The combined total number of footprints the two creatures will leave on the surfaces of their respective planets is 27,000. How many footprints does Pogo leave every meter?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the ellipse \( \frac{x^2}{4} + y^2 = 1 \) with a focus at \( F = (\sqrt{3}, 0) \), find the point \( P = (p, 0) \) such that \( p > 0 \) and for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the ellipse \( \frac{x^2}{4} + y^2 = 1 \) with a focus at \( F = (\sqrt{3}, 0) \), find the point \( P = (p, 0) \) such that \( p > 0 \) and for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal.<|im_end|>
<|im_start|>assistant
<Z_24><Z_255><Z_336><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the ellipse \( \frac{x^2}{4} + y^2 = 1 \) with a focus at \( F = (\sqrt{3}, 0) \), find the point \( P = (p, 0) \) such that \( p > 0 \) and for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal.<|im_end|>
<|im_start|>assistant
<Z_31><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the ellipse \( \frac{x^2}{4} + y^2 = 1 \) with a focus at \( F = (\sqrt{3}, 0) \), find the point \( P = (p, 0) \) such that \( p > 0 \) and for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal.<|im_end|>
<|im_start|>assistant
<Z_330><Z_83><Z_32><ANSWER>

Question: Given the ellipse \( \frac{x^2}{4} + y^2 = 1 \) with a focus at \( F = (\sqrt{3}, 0) \), find the point \( P = (p, 0) \) such that \( p > 0 \) and for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).<|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
Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).<|im_end|>
<|im_start|>assistant
<Z_369><Z_155><Z_487><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).<|im_end|>
<|im_start|>assistant
<Z_289><Z_505><Z_181><ANSWER>

Question: Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If you want to check whether a number between 1,000 and 1,100 is prime using the rule that it's enough to ensure none of the primes less than or equal to the square root of the number divide it, what is the largest prime divisor you need to test?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If you want to check whether a number between 1,000 and 1,100 is prime using the rule that it's enough to ensure none of the primes less than or equal to the square root of the number divide it, what is the largest prime divisor you need to test?<|im_end|>
<|im_start|>assistant
<Z_477><Z_227><Z_175><Z_443><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If you want to check whether a number between 1,000 and 1,100 is prime using the rule that it's enough to ensure none of the primes less than or equal to the square root of the number divide it, what is the largest prime divisor you need to test?<|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
If you want to check whether a number between 1,000 and 1,100 is prime using the rule that it's enough to ensure none of the primes less than or equal to the square root of the number divide it, what is the largest prime divisor you need to test?<|im_end|>
<|im_start|>assistant
<Z_292><Z_440><Z_256><Z_418><ANSWER>

Question: If you want to check whether a number between 1,000 and 1,100 is prime using the rule that it's enough to ensure none of the primes less than or equal to the square root of the number divide it, what is the largest prime divisor you need to test?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the positive difference between the two solutions of $|x + 3| = 15$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the positive difference between the two solutions of $|x + 3| = 15$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the positive difference between the two solutions of $|x + 3| = 15$?<|im_end|>
<|im_start|>assistant
<Z_169><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the positive difference between the two solutions of $|x + 3| = 15$?<|im_end|>
<|im_start|>assistant
<Z_234><ANSWER>

Question: What is the positive difference between the two solutions of $|x + 3| = 15$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby received his weekly paycheck today.  Although his salary is $450 per week, his actual paycheck is reduced in size because there are some taxes, insurance payments, and fees removed from his pay before his employer writes the check.  If the government removes 1/3 in federal taxes and 8% in state taxes, the health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking, then what is the final amount left in Bobby's paycheck, in dollars?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby received his weekly paycheck today.  Although his salary is $450 per week, his actual paycheck is reduced in size because there are some taxes, insurance payments, and fees removed from his pay before his employer writes the check.  If the government removes 1/3 in federal taxes and 8% in state taxes, the health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking, then what is the final amount left in Bobby's paycheck, in dollars?<|im_end|>
<|im_start|>assistant
<Z_24><Z_159><Z_116><Z_368><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby received his weekly paycheck today.  Although his salary is $450 per week, his actual paycheck is reduced in size because there are some taxes, insurance payments, and fees removed from his pay before his employer writes the check.  If the government removes 1/3 in federal taxes and 8% in state taxes, the health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking, then what is the final amount left in Bobby's paycheck, in dollars?<|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
Bobby received his weekly paycheck today.  Although his salary is $450 per week, his actual paycheck is reduced in size because there are some taxes, insurance payments, and fees removed from his pay before his employer writes the check.  If the government removes 1/3 in federal taxes and 8% in state taxes, the health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking, then what is the final amount left in Bobby's paycheck, in dollars?<|im_end|>
<|im_start|>assistant
<Z_226><Z_131><Z_14><Z_129><Z_219><ANSWER>

Question: Bobby received his weekly paycheck today.  Although his salary is $450 per week, his actual paycheck is reduced in size because there are some taxes, insurance payments, and fees removed from his pay before his employer writes the check.  If the government removes 1/3 in federal taxes and 8% in state taxes, the health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking, then what is the final amount left in Bobby's paycheck, in dollars?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares are between 100 and 400?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares are between 100 and 400?<|im_end|>
<|im_start|>assistant
<Z_369><Z_228><Z_426><Z_510><Z_278><Z_117><Z_64><Z_266><Z_387><Z_129><Z_75><Z_121><Z_136><Z_196><Z_185><Z_318><Z_229><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares are between 100 and 400?<|im_end|>
<|im_start|>assistant
<Z_205><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares are between 100 and 400?<|im_end|>
<|im_start|>assistant
<Z_337><Z_216><Z_425><Z_314><Z_1><Z_352><Z_331><Z_256><Z_8><Z_410><Z_348><Z_131><Z_79><Z_270><Z_233><Z_323><Z_362><Z_294><ANSWER>

Question: How many perfect squares are between 100 and 400?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chess team has $30$ members. In the last meeting, only 18 members were present: one-third of the girls attended, but all the boys attended. How many boys are on the chess team?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chess team has $30$ members. In the last meeting, only 18 members were present: one-third of the girls attended, but all the boys attended. How many boys are on the chess team?<|im_end|>
<|im_start|>assistant
<Z_369><Z_213><Z_111><Z_197><Z_389><Z_304><Z_186><Z_414><Z_83><Z_170><Z_53><Z_100><Z_483><Z_327><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chess team has $30$ members. In the last meeting, only 18 members were present: one-third of the girls attended, but all the boys attended. How many boys are on the chess team?<|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
A chess team has $30$ members. In the last meeting, only 18 members were present: one-third of the girls attended, but all the boys attended. How many boys are on the chess team?<|im_end|>
<|im_start|>assistant
<Z_238><Z_270><Z_125><Z_378><Z_215><Z_265><Z_420><Z_21><Z_162><Z_249><Z_451><Z_143><Z_51><Z_312><ANSWER>

Question: A chess team has $30$ members. In the last meeting, only 18 members were present: one-third of the girls attended, but all the boys attended. How many boys are on the chess team?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( N \) such that \( N < 10^{2020} \), when 7 is placed at the beginning of \( N \), the resulting number is 5 times the number formed when 7 is placed at the end of \( N \). How many different values \( N \) can take?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( N \) such that \( N < 10^{2020} \), when 7 is placed at the beginning of \( N \), the resulting number is 5 times the number formed when 7 is placed at the end of \( N \). How many different values \( N \) can take?<|im_end|>
<|im_start|>assistant
<Z_369><Z_68><Z_450><Z_232><Z_140><Z_251><Z_419><Z_75><Z_54><Z_201><Z_106><Z_407><Z_214><Z_387><Z_428><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( N \) such that \( N < 10^{2020} \), when 7 is placed at the beginning of \( N \), the resulting number is 5 times the number formed when 7 is placed at the end of \( N \). How many different values \( N \) can take?<|im_end|>
<|im_start|>assistant
<Z_213><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( N \) such that \( N < 10^{2020} \), when 7 is placed at the beginning of \( N \), the resulting number is 5 times the number formed when 7 is placed at the end of \( N \). How many different values \( N \) can take?<|im_end|>
<|im_start|>assistant
<Z_428><Z_74><Z_42><Z_225><Z_65><Z_153><Z_356><Z_232><Z_485><Z_2><Z_18><Z_469><Z_317><Z_4><Z_267><ANSWER>

Question: Given a positive integer \( N \) such that \( N < 10^{2020} \), when 7 is placed at the beginning of \( N \), the resulting number is 5 times the number formed when 7 is placed at the end of \( N \). How many different values \( N \) can take?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jamshid and Irwin decided to make up their own version of basketball. In addition to being able to score 2-point baskets, they created ways to score 5-, 11-, and 13-point baskets. Irwin scored 2 of the 11-point baskets. What was the product of the point values of Irwin’s baskets at halftime?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jamshid and Irwin decided to make up their own version of basketball. In addition to being able to score 2-point baskets, they created ways to score 5-, 11-, and 13-point baskets. Irwin scored 2 of the 11-point baskets. What was the product of the point values of Irwin’s baskets at halftime?<|im_end|>
<|im_start|>assistant
<Z_369><Z_101><Z_465><Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jamshid and Irwin decided to make up their own version of basketball. In addition to being able to score 2-point baskets, they created ways to score 5-, 11-, and 13-point baskets. Irwin scored 2 of the 11-point baskets. What was the product of the point values of Irwin’s baskets at halftime?<|im_end|>
<|im_start|>assistant
<Z_161><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jamshid and Irwin decided to make up their own version of basketball. In addition to being able to score 2-point baskets, they created ways to score 5-, 11-, and 13-point baskets. Irwin scored 2 of the 11-point baskets. What was the product of the point values of Irwin’s baskets at halftime?<|im_end|>
<|im_start|>assistant
<Z_131><Z_184><Z_434><Z_244><ANSWER>

Question: Jamshid and Irwin decided to make up their own version of basketball. In addition to being able to score 2-point baskets, they created ways to score 5-, 11-, and 13-point baskets. Irwin scored 2 of the 11-point baskets. What was the product of the point values of Irwin’s baskets at halftime?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest possible value of $k$ such that $2^{14}$ can be expressed as the sum of $k$ consecutive positive integers.<|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
Determine the largest possible value of $k$ such that $2^{14}$ can be expressed as the sum of $k$ consecutive positive integers.<|im_end|>
<|im_start|>assistant
<Z_477><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest possible value of $k$ such that $2^{14}$ can be expressed as the sum of $k$ consecutive positive integers.<|im_end|>
<|im_start|>assistant
<Z_410><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest possible value of $k$ such that $2^{14}$ can be expressed as the sum of $k$ consecutive positive integers.<|im_end|>
<|im_start|>assistant
<Z_187><Z_454><ANSWER>

Question: Determine the largest possible value of $k$ such that $2^{14}$ can be expressed as the sum of $k$ consecutive positive integers.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n = 2008^n + t$ (where $t$ is a constant), then the value of $a_1$ is.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n = 2008^n + t$ (where $t$ is a constant), then the value of $a_1$ is.<|im_end|>
<|im_start|>assistant
<Z_477><Z_311><Z_309><Z_130><Z_502><Z_451><Z_232><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n = 2008^n + t$ (where $t$ is a constant), then the value of $a_1$ is.<|im_end|>
<|im_start|>assistant
<Z_329><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n = 2008^n + t$ (where $t$ is a constant), then the value of $a_1$ is.<|im_end|>
<|im_start|>assistant
<Z_354><Z_47><Z_158><Z_387><Z_358><Z_291><Z_329><Z_329><ANSWER>

Question: If the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n = 2008^n + t$ (where $t$ is a constant), then the value of $a_1$ is.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $$\sqrt{\frac{1}{4}} \times \sqrt{16} - \left(\sqrt{\frac{1}{9}}\right)^{-1} - \sqrt{0} + \frac{\sqrt{45}}{\sqrt{5}}.$$<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $$\sqrt{\frac{1}{4}} \times \sqrt{16} - \left(\sqrt{\frac{1}{9}}\right)^{-1} - \sqrt{0} + \frac{\sqrt{45}}{\sqrt{5}}.$$<|im_end|>
<|im_start|>assistant
<Z_369><Z_372><Z_68><Z_450><Z_371><Z_351><Z_63><Z_166><Z_289><Z_99><Z_185><Z_98><Z_310><Z_64><Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $$\sqrt{\frac{1}{4}} \times \sqrt{16} - \left(\sqrt{\frac{1}{9}}\right)^{-1} - \sqrt{0} + \frac{\sqrt{45}}{\sqrt{5}}.$$<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $$\sqrt{\frac{1}{4}} \times \sqrt{16} - \left(\sqrt{\frac{1}{9}}\right)^{-1} - \sqrt{0} + \frac{\sqrt{45}}{\sqrt{5}}.$$<|im_end|>
<|im_start|>assistant
<Z_203><Z_117><Z_11><Z_366><Z_138><Z_458><Z_72><Z_9><Z_399><Z_387><Z_94><Z_338><Z_95><Z_58><Z_219><ANSWER>

Question: Calculate: $$\sqrt{\frac{1}{4}} \times \sqrt{16} - \left(\sqrt{\frac{1}{9}}\right)^{-1} - \sqrt{0} + \frac{\sqrt{45}}{\sqrt{5}}.$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
12 men can complete a piece of work in 25 days. If a larger group of men work on the same piece, they can complete it in 15 days. How many men are in the larger group?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
12 men can complete a piece of work in 25 days. If a larger group of men work on the same piece, they can complete it in 15 days. How many men are in the larger group?<|im_end|>
<|im_start|>assistant
<Z_24><Z_106><Z_465><Z_236><Z_267><Z_446><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
12 men can complete a piece of work in 25 days. If a larger group of men work on the same piece, they can complete it in 15 days. How many men are in the larger group?<|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
12 men can complete a piece of work in 25 days. If a larger group of men work on the same piece, they can complete it in 15 days. How many men are in the larger group?<|im_end|>
<|im_start|>assistant
<Z_243><Z_417><Z_402><Z_92><Z_400><Z_397><ANSWER>

Question: 12 men can complete a piece of work in 25 days. If a larger group of men work on the same piece, they can complete it in 15 days. How many men are in the larger group?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)?\]<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)?\]<|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 terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)?\]<|im_end|>
<|im_start|>assistant
<Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)?\]<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

Question: How many terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)?\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the equation log x n + log 12 n, the 3 digit number n that makes it a whole number is 1728. What is the value of x?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the equation log x n + log 12 n, the 3 digit number n that makes it a whole number is 1728. What is the value of x?<|im_end|>
<|im_start|>assistant
<Z_24><Z_364><Z_449><Z_47><Z_278><Z_159><Z_313><Z_196><Z_59><Z_445><Z_59><Z_149><Z_102><Z_68><Z_269><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the equation log x n + log 12 n, the 3 digit number n that makes it a whole number is 1728. What is the value of x?<|im_end|>
<|im_start|>assistant
<Z_277><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the equation log x n + log 12 n, the 3 digit number n that makes it a whole number is 1728. What is the value of x?<|im_end|>
<|im_start|>assistant
<Z_460><Z_373><Z_440><Z_33><Z_486><Z_48><Z_59><Z_321><Z_124><Z_506><Z_130><Z_89><Z_149><Z_484><Z_82><ANSWER>

Question: For the equation log x n + log 12 n, the 3 digit number n that makes it a whole number is 1728. What is the value of x?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, 6 social studies problems, and some science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many science problems does he have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, 6 social studies problems, and some science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many science problems does he have?<|im_end|>
<|im_start|>assistant
<Z_24><Z_386><Z_240><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, 6 social studies problems, and some science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many science problems does he have?<|im_end|>
<|im_start|>assistant
<Z_25><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, 6 social studies problems, and some science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many science problems does he have?<|im_end|>
<|im_start|>assistant
<Z_292><Z_146><Z_467><ANSWER>

Question: Brooke has 15 math problems, 6 social studies problems, and some science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many science problems does he have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A solid right cylinder with a radius of 3 feet has a total surface area of $30\pi$ square feet. What is the height of this cylinder?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A solid right cylinder with a radius of 3 feet has a total surface area of $30\pi$ square feet. What is the height of this cylinder?<|im_end|>
<|im_start|>assistant
<Z_369><Z_352><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A solid right cylinder with a radius of 3 feet has a total surface area of $30\pi$ square feet. What is the height of this cylinder?<|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
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A solid right cylinder with a radius of 3 feet has a total surface area of $30\pi$ square feet. What is the height of this cylinder?<|im_end|>
<|im_start|>assistant
<Z_307><Z_149><ANSWER>

Question: The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A solid right cylinder with a radius of 3 feet has a total surface area of $30\pi$ square feet. What is the height of this cylinder?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circle graph shows how the Megatech Corporation allocates its research and development budget: 9% microphotonics; 14% home electronics; 10% food additives; 29% genetically modified microorganisms; 8% industrial lubricants; and the remainder for basic astrophysics. If the arc of each sector of the graph is proportional to the percentage of the budget it represents, how many degrees of the circle are used to represent basic astrophysics research?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circle graph shows how the Megatech Corporation allocates its research and development budget: 9% microphotonics; 14% home electronics; 10% food additives; 29% genetically modified microorganisms; 8% industrial lubricants; and the remainder for basic astrophysics. If the arc of each sector of the graph is proportional to the percentage of the budget it represents, how many degrees of the circle are used to represent basic astrophysics research?<|im_end|>
<|im_start|>assistant
<Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circle graph shows how the Megatech Corporation allocates its research and development budget: 9% microphotonics; 14% home electronics; 10% food additives; 29% genetically modified microorganisms; 8% industrial lubricants; and the remainder for basic astrophysics. If the arc of each sector of the graph is proportional to the percentage of the budget it represents, how many degrees of the circle are used to represent basic astrophysics research?<|im_end|>
<|im_start|>assistant
<Z_434><ANSWER>

Question: A circle graph shows how the Megatech Corporation allocates its research and development budget: 9% microphotonics; 14% home electronics; 10% food additives; 29% genetically modified microorganisms; 8% industrial lubricants; and the remainder for basic astrophysics. If the arc of each sector of the graph is proportional to the percentage of the budget it represents, how many degrees of the circle are used to represent basic astrophysics research?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it?<|im_end|>
<|im_start|>assistant
<Z_477><Z_262><Z_75><Z_450><Z_217><Z_268><Z_118><Z_273><Z_432><Z_194><Z_327><Z_443><Z_246><Z_433><Z_164><Z_430><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it?<|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
How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it?<|im_end|>
<|im_start|>assistant
<Z_120><Z_258><Z_116><Z_385><Z_395><Z_430><Z_58><Z_430><Z_365><Z_391><Z_12><Z_381><Z_294><Z_35><Z_323><Z_69><ANSWER>

Question: How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the parabola $y=x^{2}+2mx-\frac{5}{4}{m^2}(m \gt 0)$
$(1)$ Prove that the parabola intersects the $x$-axis at two points;
$(2)$ If the parabola intersects the $x$-axis at points $A$ and $B$ (point $A$ is to the left of point $B$), and $AB=6$, find the value of $m$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the parabola $y=x^{2}+2mx-\frac{5}{4}{m^2}(m \gt 0)$
$(1)$ Prove that the parabola intersects the $x$-axis at two points;
$(2)$ If the parabola intersects the $x$-axis at points $A$ and $B$ (point $A$ is to the left of point $B$), and $AB=6$, find the value of $m$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_131><Z_307><Z_31><Z_87><Z_271><Z_264><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the parabola $y=x^{2}+2mx-\frac{5}{4}{m^2}(m \gt 0)$
$(1)$ Prove that the parabola intersects the $x$-axis at two points;
$(2)$ If the parabola intersects the $x$-axis at points $A$ and $B$ (point $A$ is to the left of point $B$), and $AB=6$, find the value of $m$.<|im_end|>
<|im_start|>assistant
<Z_459><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the parabola $y=x^{2}+2mx-\frac{5}{4}{m^2}(m \gt 0)$
$(1)$ Prove that the parabola intersects the $x$-axis at two points;
$(2)$ If the parabola intersects the $x$-axis at points $A$ and $B$ (point $A$ is to the left of point $B$), and $AB=6$, find the value of $m$.<|im_end|>
<|im_start|>assistant
<Z_187><Z_328><Z_120><Z_150><Z_28><Z_264><Z_278><Z_61><ANSWER>

Question:
Given the parabola $y=x^{2}+2mx-\frac{5}{4}{m^2}(m \gt 0)$
$(1)$ Prove that the parabola intersects the $x$-axis at two points;
$(2)$ If the parabola intersects the $x$-axis at points $A$ and $B$ (point $A$ is to the left of point $B$), and $AB=6$, find the value of $m$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some faces of a storage box to be painted with different colors. There are 720 ways to paint the faces with different colors. How many faces does the storage box have?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some faces of a storage box to be painted with different colors. There are 720 ways to paint the faces with different colors. How many faces does the storage box have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_418><Z_328><Z_215><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some faces of a storage box to be painted with different colors. There are 720 ways to paint the faces with different colors. How many faces does the storage box have?<|im_end|>
<|im_start|>assistant
<Z_0><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some faces of a storage box to be painted with different colors. There are 720 ways to paint the faces with different colors. How many faces does the storage box have?<|im_end|>
<|im_start|>assistant
<Z_304><Z_292><Z_310><Z_425><ANSWER>

Question: There are some faces of a storage box to be painted with different colors. There are 720 ways to paint the faces with different colors. How many faces does the storage box have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The function \( f(k) \) is a strictly increasing function defined on \( N \) and takes values in \( N \). If for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), there is \( f(x_1) < f(x_2) \), then \( f(x) \) is called a strictly increasing function on \( A \). The function satisfies the condition \( f(f(k)) = 3k \). Find \( f(1) + f(9) + f(96) \).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The function \( f(k) \) is a strictly increasing function defined on \( N \) and takes values in \( N \). If for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), there is \( f(x_1) < f(x_2) \), then \( f(x) \) is called a strictly increasing function on \( A \). The function satisfies the condition \( f(f(k)) = 3k \). Find \( f(1) + f(9) + f(96) \).<|im_end|>
<|im_start|>assistant
<Z_369><Z_310><Z_169><Z_327><Z_460><Z_321><Z_140><Z_352><Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The function \( f(k) \) is a strictly increasing function defined on \( N \) and takes values in \( N \). If for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), there is \( f(x_1) < f(x_2) \), then \( f(x) \) is called a strictly increasing function on \( A \). The function satisfies the condition \( f(f(k)) = 3k \). Find \( f(1) + f(9) + f(96) \).<|im_end|>
<|im_start|>assistant
<Z_131><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The function \( f(k) \) is a strictly increasing function defined on \( N \) and takes values in \( N \). If for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), there is \( f(x_1) < f(x_2) \), then \( f(x) \) is called a strictly increasing function on \( A \). The function satisfies the condition \( f(f(k)) = 3k \). Find \( f(1) + f(9) + f(96) \).<|im_end|>
<|im_start|>assistant
<Z_195><Z_85><Z_74><Z_269><Z_83><Z_144><Z_41><Z_58><Z_393><ANSWER>

Question: The function \( f(k) \) is a strictly increasing function defined on \( N \) and takes values in \( N \). If for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), there is \( f(x_1) < f(x_2) \), then \( f(x) \) is called a strictly increasing function on \( A \). The function satisfies the condition \( f(f(k)) = 3k \). Find \( f(1) + f(9) + f(96) \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $b$ is an odd multiple of $8723$, find the greatest common divisor of $8b^2 + 55b + 144$ and $4b + 15$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $b$ is an odd multiple of $8723$, find the greatest common divisor of $8b^2 + 55b + 144$ and $4b + 15$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_504><Z_465><Z_144><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $b$ is an odd multiple of $8723$, find the greatest common divisor of $8b^2 + 55b + 144$ and $4b + 15$.<|im_end|>
<|im_start|>assistant
<Z_95><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $b$ is an odd multiple of $8723$, find the greatest common divisor of $8b^2 + 55b + 144$ and $4b + 15$.<|im_end|>
<|im_start|>assistant
<Z_464><Z_122><Z_315><Z_236><ANSWER>

Question: Given that $b$ is an odd multiple of $8723$, find the greatest common divisor of $8b^2 + 55b + 144$ and $4b + 15$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain number plus 16 2/3% of another number equals 10500. The other number is 9032. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain number plus 16 2/3% of another number equals 10500. The other number is 9032. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_369><Z_223><Z_470><Z_181><Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain number plus 16 2/3% of another number equals 10500. The other number is 9032. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_23><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain number plus 16 2/3% of another number equals 10500. The other number is 9032. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_62><Z_63><Z_143><Z_293><Z_102><ANSWER>

Question: A certain number plus 16 2/3% of another number equals 10500. The other number is 9032. What is the first number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Show that if \( p \) is a prime number, then \( p \) divides \( (p-1)! + 1 \). Show that if \( n \) is not a prime number, then \( n \) does not divide \( (n-1)! + 1 \). Calculate the remainder of the division of \( (n-1)! \) by \( n \).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Show that if \( p \) is a prime number, then \( p \) divides \( (p-1)! + 1 \). Show that if \( n \) is not a prime number, then \( n \) does not divide \( (n-1)! + 1 \). Calculate the remainder of the division of \( (n-1)! \) by \( n \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_288><Z_146><Z_181><Z_436><Z_462><Z_357><Z_437><Z_77><Z_304><Z_378><Z_396><Z_443><Z_465><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Show that if \( p \) is a prime number, then \( p \) divides \( (p-1)! + 1 \). Show that if \( n \) is not a prime number, then \( n \) does not divide \( (n-1)! + 1 \). Calculate the remainder of the division of \( (n-1)! \) by \( n \).<|im_end|>
<|im_start|>assistant
<Z_415><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Show that if \( p \) is a prime number, then \( p \) divides \( (p-1)! + 1 \). Show that if \( n \) is not a prime number, then \( n \) does not divide \( (n-1)! + 1 \). Calculate the remainder of the division of \( (n-1)! \) by \( n \).<|im_end|>
<|im_start|>assistant
<Z_339><Z_98><Z_333><Z_507><Z_5><Z_11><Z_94><Z_361><Z_288><Z_245><Z_35><Z_452><Z_62><Z_485><Z_191><ANSWER>

Question: Show that if \( p \) is a prime number, then \( p \) divides \( (p-1)! + 1 \). Show that if \( n \) is not a prime number, then \( n \) does not divide \( (n-1)! + 1 \). Calculate the remainder of the division of \( (n-1)! \) by \( n \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?<|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
In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?<|im_end|>
<|im_start|>assistant
<Z_117><ANSWER>

Question: In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sports club with 42 members, 20 play badminton and 23 play tennis. There are 7 members who play both badminton and tennis. How many members do not play either badminton or tennis?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sports club with 42 members, 20 play badminton and 23 play tennis. There are 7 members who play both badminton and tennis. How many members do not play either badminton or tennis?<|im_end|>
<|im_start|>assistant
<Z_24><Z_386><Z_102><Z_206><Z_362><Z_425><Z_116><Z_321><Z_107><Z_89><Z_460><Z_52><Z_362><Z_439><Z_174><Z_362><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sports club with 42 members, 20 play badminton and 23 play tennis. There are 7 members who play both badminton and tennis. How many members do not play either badminton or tennis?<|im_end|>
<|im_start|>assistant
<Z_329><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sports club with 42 members, 20 play badminton and 23 play tennis. There are 7 members who play both badminton and tennis. How many members do not play either badminton or tennis?<|im_end|>
<|im_start|>assistant
<Z_2><Z_205><Z_484><Z_394><Z_32><Z_148><Z_423><Z_241><Z_366><Z_158><Z_280><Z_61><Z_465><Z_276><Z_272><Z_120><ANSWER>

Question: In a sports club with 42 members, 20 play badminton and 23 play tennis. There are 7 members who play both badminton and tennis. How many members do not play either badminton or tennis?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[P = 123321\] and \[Q = 246642\]. Find a positive integer $M$ such that $M$ is the multiplicative inverse of $PQ$ modulo 1,000,003.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[P = 123321\] and \[Q = 246642\]. Find a positive integer $M$ such that $M$ is the multiplicative inverse of $PQ$ modulo 1,000,003.<|im_end|>
<|im_start|>assistant
<Z_24><Z_441><Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[P = 123321\] and \[Q = 246642\]. Find a positive integer $M$ such that $M$ is the multiplicative inverse of $PQ$ modulo 1,000,003.<|im_end|>
<|im_start|>assistant
<Z_455><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[P = 123321\] and \[Q = 246642\]. Find a positive integer $M$ such that $M$ is the multiplicative inverse of $PQ$ modulo 1,000,003.<|im_end|>
<|im_start|>assistant
<Z_396><Z_493><Z_247><ANSWER>

Question: Let \[P = 123321\] and \[Q = 246642\]. Find a positive integer $M$ such that $M$ is the multiplicative inverse of $PQ$ modulo 1,000,003.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make?<|im_end|>
<|im_start|>assistant
<Z_24><Z_89><Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make?<|im_end|>
<|im_start|>assistant
<Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make?<|im_end|>
<|im_start|>assistant
<Z_57><Z_177><Z_485><ANSWER>

Question: A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|im_end|>
<|im_start|>assistant
<Z_369><Z_155><Z_421><Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|im_end|>
<|im_start|>assistant
<Z_404><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|im_end|>
<|im_start|>assistant
<Z_23><Z_2><Z_250><Z_257><ANSWER>

Question: Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|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
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|im_end|>
<|im_start|>assistant
<Z_24><Z_378><Z_91><Z_342><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|im_end|>
<|im_start|>assistant
<Z_146><Z_196><Z_109><Z_1><ANSWER>

Question: Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|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
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_143><ANSWER>

Question: Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?