<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three planes are going to the same place but each has a different number of passengers. The first plane has 50, the second had 60, and the third has some passengers. An empty plane can go 600 MPH, but each passenger makes it go 2 MPH slower. Their average speed is 500 MPH. How many passengers are on the third plane?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three planes are going to the same place but each has a different number of passengers. The first plane has 50, the second had 60, and the third has some passengers. An empty plane can go 600 MPH, but each passenger makes it go 2 MPH slower. Their average speed is 500 MPH. How many passengers are on the third plane?<|im_end|>
<|im_start|>assistant
<Z_358><Z_503><Z_395><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three planes are going to the same place but each has a different number of passengers. The first plane has 50, the second had 60, and the third has some passengers. An empty plane can go 600 MPH, but each passenger makes it go 2 MPH slower. Their average speed is 500 MPH. How many passengers are on the third plane?<|im_end|>
<|im_start|>assistant
<Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three planes are going to the same place but each has a different number of passengers. The first plane has 50, the second had 60, and the third has some passengers. An empty plane can go 600 MPH, but each passenger makes it go 2 MPH slower. Their average speed is 500 MPH. How many passengers are on the third plane?<|im_end|>
<|im_start|>assistant
<Z_353><Z_117><Z_182><ANSWER>

Question: Three planes are going to the same place but each has a different number of passengers. The first plane has 50, the second had 60, and the third has some passengers. An empty plane can go 600 MPH, but each passenger makes it go 2 MPH slower. Their average speed is 500 MPH. How many passengers are on the third plane?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an office, the average salary of officers is Rs. 450 and of non-officers is Rs. 110. There are 15 officers and 495 non-officers. What is the average salary of all employees in the office?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an office, the average salary of officers is Rs. 450 and of non-officers is Rs. 110. There are 15 officers and 495 non-officers. What is the average salary of all employees in the office?<|im_end|>
<|im_start|>assistant
<Z_176><Z_202><Z_43><Z_421><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an office, the average salary of officers is Rs. 450 and of non-officers is Rs. 110. There are 15 officers and 495 non-officers. What is the average salary of all employees in the office?<|im_end|>
<|im_start|>assistant
<Z_327><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an office, the average salary of officers is Rs. 450 and of non-officers is Rs. 110. There are 15 officers and 495 non-officers. What is the average salary of all employees in the office?<|im_end|>
<|im_start|>assistant
<Z_302><Z_504><Z_118><Z_362><ANSWER>

Question: In an office, the average salary of officers is Rs. 450 and of non-officers is Rs. 110. There are 15 officers and 495 non-officers. What is the average salary of all employees in the office?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A group of 30 tourists sets out on a walking tour of the rainforest. Two tourists are eaten by anacondas, then half the remaining tourists try to pet poisonous dart frogs and get poisoned. If only 1/7 of the poisoned tourists recovered, how many tourists are left at the end of the tour?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A group of 30 tourists sets out on a walking tour of the rainforest. Two tourists are eaten by anacondas, then half the remaining tourists try to pet poisonous dart frogs and get poisoned. If only 1/7 of the poisoned tourists recovered, how many tourists are left at the end of the tour?<|im_end|>
<|im_start|>assistant
<Z_358><Z_467><Z_78><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A group of 30 tourists sets out on a walking tour of the rainforest. Two tourists are eaten by anacondas, then half the remaining tourists try to pet poisonous dart frogs and get poisoned. If only 1/7 of the poisoned tourists recovered, how many tourists are left at the end of the tour?<|im_end|>
<|im_start|>assistant
<Z_294><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A group of 30 tourists sets out on a walking tour of the rainforest. Two tourists are eaten by anacondas, then half the remaining tourists try to pet poisonous dart frogs and get poisoned. If only 1/7 of the poisoned tourists recovered, how many tourists are left at the end of the tour?<|im_end|>
<|im_start|>assistant
<Z_203><Z_176><Z_510><Z_358><ANSWER>

Question: A group of 30 tourists sets out on a walking tour of the rainforest. Two tourists are eaten by anacondas, then half the remaining tourists try to pet poisonous dart frogs and get poisoned. If only 1/7 of the poisoned tourists recovered, how many tourists are left at the end of the tour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest natural number \( n \) with the following property: for any odd prime \( p \) less than \( n \), the difference \( n - p \) is also a prime number.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest natural number \( n \) with the following property: for any odd prime \( p \) less than \( n \), the difference \( n - p \) is also a prime number.<|im_end|>
<|im_start|>assistant
<Z_66><Z_147><Z_477><Z_442><Z_122><Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest natural number \( n \) with the following property: for any odd prime \( p \) less than \( n \), the difference \( n - p \) is also a prime number.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest natural number \( n \) with the following property: for any odd prime \( p \) less than \( n \), the difference \( n - p \) is also a prime number.<|im_end|>
<|im_start|>assistant
<Z_467><Z_170><Z_489><Z_384><Z_483><Z_310><ANSWER>

Question: Find the largest natural number \( n \) with the following property: for any odd prime \( p \) less than \( n \), the difference \( n - p \) is also a prime number.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .

*Romania*<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .

*Romania*<|im_end|>
<|im_start|>assistant
<Z_176><Z_386><Z_461><Z_276><Z_453><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .

*Romania*<|im_end|>
<|im_start|>assistant
<Z_192><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .

*Romania*<|im_end|>
<|im_start|>assistant
<Z_9><Z_195><Z_456><Z_426><Z_253><ANSWER>

Question:
For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .

*Romania*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A regular polygon has interior angles of 108 degrees. How many sides does the polygon have?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A regular polygon has interior angles of 108 degrees. How many sides does the polygon have?<|im_end|>
<|im_start|>assistant
<Z_5><Z_499><Z_429><Z_303><Z_439><Z_140><Z_93><Z_225><Z_295><Z_360><Z_391><Z_337><Z_507><Z_310><Z_330><Z_168><Z_454><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A regular polygon has interior angles of 108 degrees. How many sides does the polygon have?<|im_end|>
<|im_start|>assistant
<Z_237><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A regular polygon has interior angles of 108 degrees. How many sides does the polygon have?<|im_end|>
<|im_start|>assistant
<Z_135><Z_177><Z_6><Z_277><Z_310><Z_364><Z_204><Z_55><Z_478><Z_371><Z_404><Z_404><Z_441><Z_428><Z_143><Z_6><Z_503><Z_404><ANSWER>

Question: A regular polygon has interior angles of 108 degrees. How many sides does the polygon have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a square-shaped piece of cardboard is 28 dm. How many square centimeters is its area?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a square-shaped piece of cardboard is 28 dm. How many square centimeters is its area?<|im_end|>
<|im_start|>assistant
<Z_358><Z_300><Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a square-shaped piece of cardboard is 28 dm. How many square centimeters is its area?<|im_end|>
<|im_start|>assistant
<Z_152><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a square-shaped piece of cardboard is 28 dm. How many square centimeters is its area?<|im_end|>
<|im_start|>assistant
<Z_81><Z_338><Z_247><ANSWER>

Question: The perimeter of a square-shaped piece of cardboard is 28 dm. How many square centimeters is its area?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $78$ and $182$?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $78$ and $182$?<|im_end|>
<|im_start|>assistant
<Z_195><Z_452><Z_133><Z_33><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 common divisor of $78$ and $182$?<|im_end|>
<|im_start|>assistant
<Z_149><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $78$ and $182$?<|im_end|>
<|im_start|>assistant
<Z_263><Z_187><Z_233><Z_473><ANSWER>

Question: What is the greatest common divisor of $78$ and $182$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the arithmetic sequence $\{a_n\}$, we have $a_{10}=33$ and $a_2=1$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Compute the value of $S_{20}-2S_{10}$.<|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
For the arithmetic sequence $\{a_n\}$, we have $a_{10}=33$ and $a_2=1$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Compute the value of $S_{20}-2S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_189><Z_295><Z_501><Z_124><Z_231><Z_260><Z_223><Z_275><Z_488><Z_425><Z_340><Z_387><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the arithmetic sequence $\{a_n\}$, we have $a_{10}=33$ and $a_2=1$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Compute the value of $S_{20}-2S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the arithmetic sequence $\{a_n\}$, we have $a_{10}=33$ and $a_2=1$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Compute the value of $S_{20}-2S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_474><Z_282><Z_157><Z_506><Z_392><Z_161><Z_16><Z_471><Z_265><Z_59><Z_78><Z_151><Z_192><Z_27><ANSWER>

Question: For the arithmetic sequence $\{a_n\}$, we have $a_{10}=33$ and $a_2=1$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Compute the value of $S_{20}-2S_{10}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = |\log_{10} x|$, if there exist distinct real numbers $a$ and $b$ such that $f(a) = f(b)$, find the value of $ab$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = |\log_{10} x|$, if there exist distinct real numbers $a$ and $b$ such that $f(a) = f(b)$, find the value of $ab$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_74><Z_56><Z_269><Z_383><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = |\log_{10} x|$, if there exist distinct real numbers $a$ and $b$ such that $f(a) = f(b)$, find the value of $ab$.<|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
Given the function $f(x) = |\log_{10} x|$, if there exist distinct real numbers $a$ and $b$ such that $f(a) = f(b)$, find the value of $ab$.<|im_end|>
<|im_start|>assistant
<Z_415><Z_81><Z_445><Z_71><Z_26><ANSWER>

Question: Given the function $f(x) = |\log_{10} x|$, if there exist distinct real numbers $a$ and $b$ such that $f(a) = f(b)$, find the value of $ab$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the radius of the circle with equation $2x^2 - 8x + 2y^2 + 4y + 10 = 0$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the radius of the circle with equation $2x^2 - 8x + 2y^2 + 4y + 10 = 0$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_161><Z_510><Z_150><Z_500><Z_301><Z_228><Z_489><Z_351><Z_369><Z_43><Z_253><Z_62><Z_278><Z_313><Z_209><Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the radius of the circle with equation $2x^2 - 8x + 2y^2 + 4y + 10 = 0$.<|im_end|>
<|im_start|>assistant
<Z_222><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the radius of the circle with equation $2x^2 - 8x + 2y^2 + 4y + 10 = 0$.<|im_end|>
<|im_start|>assistant
<Z_48><Z_151><Z_293><Z_468><Z_272><Z_465><Z_193><Z_49><Z_461><Z_86><Z_435><Z_387><Z_248><Z_157><Z_346><Z_269><Z_186><ANSWER>

Question: Find the radius of the circle with equation $2x^2 - 8x + 2y^2 + 4y + 10 = 0$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.<|im_end|>
<|im_start|>assistant
<Z_5><Z_75><Z_409><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.<|im_end|>
<|im_start|>assistant
<Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.<|im_end|>
<|im_start|>assistant
<Z_111><Z_461><Z_410><ANSWER>

Question: Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 9 students in the class and a certain number of Skittles. If the Skittles are divided equally among the students, each student gets 3 Skittles. How many Skittles are there in total?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 9 students in the class and a certain number of Skittles. If the Skittles are divided equally among the students, each student gets 3 Skittles. How many Skittles are there in total?<|im_end|>
<|im_start|>assistant
<Z_195><Z_244><Z_141><Z_359><Z_181><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 9 students in the class and a certain number of Skittles. If the Skittles are divided equally among the students, each student gets 3 Skittles. How many Skittles are there in total?<|im_end|>
<|im_start|>assistant
<Z_85><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 9 students in the class and a certain number of Skittles. If the Skittles are divided equally among the students, each student gets 3 Skittles. How many Skittles are there in total?<|im_end|>
<|im_start|>assistant
<Z_328><Z_309><Z_243><Z_160><Z_209><ANSWER>

Question: There are 9 students in the class and a certain number of Skittles. If the Skittles are divided equally among the students, each student gets 3 Skittles. How many Skittles are there in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car travels 69 m in 3 seconds, how fast is it going?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car travels 69 m in 3 seconds, how fast is it going?<|im_end|>
<|im_start|>assistant
<Z_195><Z_254><Z_370><Z_126><Z_48><Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car travels 69 m in 3 seconds, how fast is it going?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car travels 69 m in 3 seconds, how fast is it going?<|im_end|>
<|im_start|>assistant
<Z_61><Z_252><Z_151><Z_320><Z_150><Z_337><ANSWER>

Question: If a car travels 69 m in 3 seconds, how fast is it going?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four runners ran a combined total of some miles last week. Katarina ran 51 miles. Tomas, Tyler, and Harriet all ran the same distance, and Harriet ran 48 miles. What was the combined total of miles run by all four runners?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four runners ran a combined total of some miles last week. Katarina ran 51 miles. Tomas, Tyler, and Harriet all ran the same distance, and Harriet ran 48 miles. What was the combined total of miles run by all four runners?<|im_end|>
<|im_start|>assistant
<Z_5><Z_339><Z_376><Z_70><Z_453><Z_435><Z_153><Z_300><Z_265><Z_274><Z_100><Z_127><Z_406><Z_304><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four runners ran a combined total of some miles last week. Katarina ran 51 miles. Tomas, Tyler, and Harriet all ran the same distance, and Harriet ran 48 miles. What was the combined total of miles run by all four runners?<|im_end|>
<|im_start|>assistant
<Z_484><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four runners ran a combined total of some miles last week. Katarina ran 51 miles. Tomas, Tyler, and Harriet all ran the same distance, and Harriet ran 48 miles. What was the combined total of miles run by all four runners?<|im_end|>
<|im_start|>assistant
<Z_86><Z_1><Z_221><Z_8><Z_185><Z_399><Z_450><Z_501><Z_300><Z_156><Z_218><Z_299><Z_79><Z_240><ANSWER>

Question: Four runners ran a combined total of some miles last week. Katarina ran 51 miles. Tomas, Tyler, and Harriet all ran the same distance, and Harriet ran 48 miles. What was the combined total of miles run by all four runners?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\).<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\).<|im_end|>
<|im_start|>assistant
<Z_5><Z_372><Z_396><Z_373><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\).<|im_end|>
<|im_start|>assistant
<Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\).<|im_end|>
<|im_start|>assistant
<Z_100><Z_327><Z_7><Z_88><ANSWER>

Question: Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jed collects stamp cards. Every week, he gets 6 cards. But every two weeks, he gives 2 cards to his friends. If Jed started with 20 cards, after how many weeks will he have a total of 40 cards?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jed collects stamp cards. Every week, he gets 6 cards. But every two weeks, he gives 2 cards to his friends. If Jed started with 20 cards, after how many weeks will he have a total of 40 cards?<|im_end|>
<|im_start|>assistant
<Z_176><Z_53><Z_128><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jed collects stamp cards. Every week, he gets 6 cards. But every two weeks, he gives 2 cards to his friends. If Jed started with 20 cards, after how many weeks will he have a total of 40 cards?<|im_end|>
<|im_start|>assistant
<Z_187><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jed collects stamp cards. Every week, he gets 6 cards. But every two weeks, he gives 2 cards to his friends. If Jed started with 20 cards, after how many weeks will he have a total of 40 cards?<|im_end|>
<|im_start|>assistant
<Z_119><Z_232><Z_200><ANSWER>

Question: Jed collects stamp cards. Every week, he gets 6 cards. But every two weeks, he gives 2 cards to his friends. If Jed started with 20 cards, after how many weeks will he have a total of 40 cards?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_176><Z_364><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_484><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_15><Z_492><ANSWER>

Question: The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In rectangle $ABCD$, $AB = 4$ and $BC = 10$. The rectangle is folded along a line through $A$ such that point $A$ meets side $CD$ at point $G$, where $DG = 3$. After the fold, point $C$ touches the extension of side $AB$ at point $E$. Calculate the length of segment $CE$. Express your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In rectangle $ABCD$, $AB = 4$ and $BC = 10$. The rectangle is folded along a line through $A$ such that point $A$ meets side $CD$ at point $G$, where $DG = 3$. After the fold, point $C$ touches the extension of side $AB$ at point $E$. Calculate the length of segment $CE$. Express your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_358><Z_476><Z_405><Z_83><Z_474><Z_368><Z_133><Z_337><Z_300><Z_202><Z_261><Z_123><Z_307><Z_389><Z_362><Z_347><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In rectangle $ABCD$, $AB = 4$ and $BC = 10$. The rectangle is folded along a line through $A$ such that point $A$ meets side $CD$ at point $G$, where $DG = 3$. After the fold, point $C$ touches the extension of side $AB$ at point $E$. Calculate the length of segment $CE$. Express your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_442><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In rectangle $ABCD$, $AB = 4$ and $BC = 10$. The rectangle is folded along a line through $A$ such that point $A$ meets side $CD$ at point $G$, where $DG = 3$. After the fold, point $C$ touches the extension of side $AB$ at point $E$. Calculate the length of segment $CE$. Express your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_181><Z_284><Z_265><Z_25><Z_340><Z_10><Z_440><Z_97><Z_87><Z_180><Z_321><Z_182><Z_397><Z_434><Z_380><Z_276><ANSWER>

Question: In rectangle $ABCD$, $AB = 4$ and $BC = 10$. The rectangle is folded along a line through $A$ such that point $A$ meets side $CD$ at point $G$, where $DG = 3$. After the fold, point $C$ touches the extension of side $AB$ at point $E$. Calculate the length of segment $CE$. Express your answer in simplest radical form.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x and y are two towns . ganesh covers the distance from x to y at an average speed of 60 km / hr . however , he covers the distance from y to x at an average speed of 30 km / hr . his average speed during the whole journey in km / hr . is :<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x and y are two towns . ganesh covers the distance from x to y at an average speed of 60 km / hr . however , he covers the distance from y to x at an average speed of 30 km / hr . his average speed during the whole journey in km / hr . is :<|im_end|>
<|im_start|>assistant
<Z_66><Z_270><Z_118><Z_256><Z_205><Z_399><Z_421><Z_100><Z_477><Z_434><Z_342><Z_305><Z_391><Z_204><Z_316><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x and y are two towns . ganesh covers the distance from x to y at an average speed of 60 km / hr . however , he covers the distance from y to x at an average speed of 30 km / hr . his average speed during the whole journey in km / hr . is :<|im_end|>
<|im_start|>assistant
<Z_271><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x and y are two towns . ganesh covers the distance from x to y at an average speed of 60 km / hr . however , he covers the distance from y to x at an average speed of 30 km / hr . his average speed during the whole journey in km / hr . is :<|im_end|>
<|im_start|>assistant
<Z_458><Z_226><Z_30><Z_180><Z_28><Z_117><Z_412><Z_501><Z_160><Z_156><Z_248><Z_177><Z_392><Z_29><Z_314><ANSWER>

Question: x and y are two towns . ganesh covers the distance from x to y at an average speed of 60 km / hr . however , he covers the distance from y to x at an average speed of 30 km / hr . his average speed during the whole journey in km / hr . is :


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many two-digit prime numbers have a units digit of 9?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many two-digit prime numbers have a units digit of 9?<|im_end|>
<|im_start|>assistant
<Z_66><Z_333><Z_341><Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many two-digit prime numbers have a units digit of 9?<|im_end|>
<|im_start|>assistant
<Z_153><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many two-digit prime numbers have a units digit of 9?<|im_end|>
<|im_start|>assistant
<Z_352><Z_428><Z_173><Z_87><ANSWER>

Question: How many two-digit prime numbers have a units digit of 9?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I have a picture with dimensions $x$ and $y$ (in inches), where $x$ and $y$ are integers greater than one. I decide to place this picture in a larger frame with dimensions $(3x + 3)$ and $(y + 2)$. If the total area of the frame is measured to be $110$ square inches, what is the area of the picture in square inches?<|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
I have a picture with dimensions $x$ and $y$ (in inches), where $x$ and $y$ are integers greater than one. I decide to place this picture in a larger frame with dimensions $(3x + 3)$ and $(y + 2)$. If the total area of the frame is measured to be $110$ square inches, what is the area of the picture in square inches?<|im_end|>
<|im_start|>assistant
<Z_195><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I have a picture with dimensions $x$ and $y$ (in inches), where $x$ and $y$ are integers greater than one. I decide to place this picture in a larger frame with dimensions $(3x + 3)$ and $(y + 2)$. If the total area of the frame is measured to be $110$ square inches, what is the area of the picture in square inches?<|im_end|>
<|im_start|>assistant
<Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I have a picture with dimensions $x$ and $y$ (in inches), where $x$ and $y$ are integers greater than one. I decide to place this picture in a larger frame with dimensions $(3x + 3)$ and $(y + 2)$. If the total area of the frame is measured to be $110$ square inches, what is the area of the picture in square inches?<|im_end|>
<|im_start|>assistant
<Z_379><Z_227><ANSWER>

Question: I have a picture with dimensions $x$ and $y$ (in inches), where $x$ and $y$ are integers greater than one. I decide to place this picture in a larger frame with dimensions $(3x + 3)$ and $(y + 2)$. If the total area of the frame is measured to be $110$ square inches, what is the area of the picture in square inches?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $f(x)$ is an even function defined on $\mathbb{R}$, let $F(x) = (x - b)f(x - b) + 1009$. If the real number $b$ satisfies $2b = a + c$, then find $F(a) + F(c) = \_\_\_\_\_\_.$<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $f(x)$ is an even function defined on $\mathbb{R}$, let $F(x) = (x - b)f(x - b) + 1009$. If the real number $b$ satisfies $2b = a + c$, then find $F(a) + F(c) = \_\_\_\_\_\_.$<|im_end|>
<|im_start|>assistant
<Z_5><Z_98><Z_153><Z_435><Z_83><Z_400><Z_341><Z_150><Z_307><Z_254><Z_84><Z_187><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $f(x)$ is an even function defined on $\mathbb{R}$, let $F(x) = (x - b)f(x - b) + 1009$. If the real number $b$ satisfies $2b = a + c$, then find $F(a) + F(c) = \_\_\_\_\_\_.$<|im_end|>
<|im_start|>assistant
<Z_370><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $f(x)$ is an even function defined on $\mathbb{R}$, let $F(x) = (x - b)f(x - b) + 1009$. If the real number $b$ satisfies $2b = a + c$, then find $F(a) + F(c) = \_\_\_\_\_\_.$<|im_end|>
<|im_start|>assistant
<Z_46><Z_190><Z_495><Z_353><Z_8><Z_203><Z_475><Z_206><Z_27><Z_448><Z_161><Z_407><ANSWER>

Question: Given that $f(x)$ is an even function defined on $\mathbb{R}$, let $F(x) = (x - b)f(x - b) + 1009$. If the real number $b$ satisfies $2b = a + c$, then find $F(a) + F(c) = \_\_\_\_\_\_.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x,$ $y,$ $z,$ $w$ be positive real numbers such that $x + y + z + w = 1.$ Find the minimum value of
\[
\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{x+w} + \frac{1}{y+z} + \frac{1}{y+w} + \frac{1}{z+w}.
\]<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x,$ $y,$ $z,$ $w$ be positive real numbers such that $x + y + z + w = 1.$ Find the minimum value of
\[
\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{x+w} + \frac{1}{y+z} + \frac{1}{y+w} + \frac{1}{z+w}.
\]<|im_end|>
<|im_start|>assistant
<Z_176><Z_75><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x,$ $y,$ $z,$ $w$ be positive real numbers such that $x + y + z + w = 1.$ Find the minimum value of
\[
\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{x+w} + \frac{1}{y+z} + \frac{1}{y+w} + \frac{1}{z+w}.
\]<|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
Let $x,$ $y,$ $z,$ $w$ be positive real numbers such that $x + y + z + w = 1.$ Find the minimum value of
\[
\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{x+w} + \frac{1}{y+z} + \frac{1}{y+w} + \frac{1}{z+w}.
\]<|im_end|>
<|im_start|>assistant
<Z_411><Z_280><Z_230><ANSWER>

Question:
Let $x,$ $y,$ $z,$ $w$ be positive real numbers such that $x + y + z + w = 1.$ Find the minimum value of
\[
\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{x+w} + \frac{1}{y+z} + \frac{1}{y+w} + \frac{1}{z+w}.
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( N \) as the number of digits of the number \( 2^{100} \), determine a positive integer \( k \) such that \( k \leq N \leq k+5 \).<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( N \) as the number of digits of the number \( 2^{100} \), determine a positive integer \( k \) such that \( k \leq N \leq k+5 \).<|im_end|>
<|im_start|>assistant
<Z_66><Z_313><Z_320><Z_341><Z_323><Z_165><Z_231><Z_449><Z_107><Z_203><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( N \) as the number of digits of the number \( 2^{100} \), determine a positive integer \( k \) such that \( k \leq N \leq k+5 \).<|im_end|>
<|im_start|>assistant
<Z_242><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( N \) as the number of digits of the number \( 2^{100} \), determine a positive integer \( k \) such that \( k \leq N \leq k+5 \).<|im_end|>
<|im_start|>assistant
<Z_343><Z_283><Z_360><Z_400><Z_501><Z_201><Z_396><Z_59><Z_254><Z_52><ANSWER>

Question: Given \( N \) as the number of digits of the number \( 2^{100} \), determine a positive integer \( k \) such that \( k \leq N \leq k+5 \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a large residential complex, there are 500 apartments. 90% of them have at least 1 resident, while 72% of the apartments have at least two residents. How many apartments have only one resident?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a large residential complex, there are 500 apartments. 90% of them have at least 1 resident, while 72% of the apartments have at least two residents. How many apartments have only one resident?<|im_end|>
<|im_start|>assistant
<Z_66><Z_389><Z_76><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a large residential complex, there are 500 apartments. 90% of them have at least 1 resident, while 72% of the apartments have at least two residents. How many apartments have only one resident?<|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
In a large residential complex, there are 500 apartments. 90% of them have at least 1 resident, while 72% of the apartments have at least two residents. How many apartments have only one resident?<|im_end|>
<|im_start|>assistant
<Z_249><Z_366><Z_285><Z_186><ANSWER>

Question: In a large residential complex, there are 500 apartments. 90% of them have at least 1 resident, while 72% of the apartments have at least two residents. How many apartments have only one resident?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at the speed of 120 km/hr crosses a pole in 15 seconds. Find the length of the train.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at the speed of 120 km/hr crosses a pole in 15 seconds. Find the length of the train.<|im_end|>
<|im_start|>assistant
<Z_238><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at the speed of 120 km/hr crosses a pole in 15 seconds. Find the length of the train.<|im_end|>
<|im_start|>assistant
<Z_134><ANSWER>

Question: A train running at the speed of 120 km/hr crosses a pole in 15 seconds. Find the length of the train.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan weighs 5 more pounds than Steve. Steve is some pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, their total weight is 319 pounds. How many pounds lighter is Steve than Jim?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan weighs 5 more pounds than Steve. Steve is some pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, their total weight is 319 pounds. How many pounds lighter is Steve than Jim?<|im_end|>
<|im_start|>assistant
<Z_176><Z_507><Z_400><Z_60><Z_407><Z_196><Z_300><Z_313><Z_147><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan weighs 5 more pounds than Steve. Steve is some pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, their total weight is 319 pounds. How many pounds lighter is Steve than Jim?<|im_end|>
<|im_start|>assistant
<Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan weighs 5 more pounds than Steve. Steve is some pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, their total weight is 319 pounds. How many pounds lighter is Steve than Jim?<|im_end|>
<|im_start|>assistant
<Z_314><Z_323><Z_78><Z_443><Z_499><Z_79><Z_11><Z_2><Z_388><ANSWER>

Question: Stan weighs 5 more pounds than Steve. Steve is some pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, their total weight is 319 pounds. How many pounds lighter is Steve than Jim?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacob has been tasked with a project to write up an urban plan proposal that will be used to construct a new city. He uses a reference city which has some stores, 500 hospitals, 200 schools, and 20 police stations. The new city should have half the number of stores, twice as many hospitals, 50 fewer schools, and 5 more police stations. The total number of buildings required for this project is 2175. How many stores does the reference city have?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacob has been tasked with a project to write up an urban plan proposal that will be used to construct a new city. He uses a reference city which has some stores, 500 hospitals, 200 schools, and 20 police stations. The new city should have half the number of stores, twice as many hospitals, 50 fewer schools, and 5 more police stations. The total number of buildings required for this project is 2175. How many stores does the reference city have?<|im_end|>
<|im_start|>assistant
<Z_195><Z_476><Z_253><Z_361><Z_392><Z_181><Z_181><Z_387><Z_446><Z_234><Z_488><Z_345><Z_323><Z_319><Z_322><Z_495><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacob has been tasked with a project to write up an urban plan proposal that will be used to construct a new city. He uses a reference city which has some stores, 500 hospitals, 200 schools, and 20 police stations. The new city should have half the number of stores, twice as many hospitals, 50 fewer schools, and 5 more police stations. The total number of buildings required for this project is 2175. How many stores does the reference city have?<|im_end|>
<|im_start|>assistant
<Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacob has been tasked with a project to write up an urban plan proposal that will be used to construct a new city. He uses a reference city which has some stores, 500 hospitals, 200 schools, and 20 police stations. The new city should have half the number of stores, twice as many hospitals, 50 fewer schools, and 5 more police stations. The total number of buildings required for this project is 2175. How many stores does the reference city have?<|im_end|>
<|im_start|>assistant
<Z_266><Z_323><Z_393><Z_195><Z_247><Z_357><Z_308><Z_263><Z_213><Z_71><Z_248><Z_49><Z_23><Z_41><Z_351><Z_57><ANSWER>

Question: Jacob has been tasked with a project to write up an urban plan proposal that will be used to construct a new city. He uses a reference city which has some stores, 500 hospitals, 200 schools, and 20 police stations. The new city should have half the number of stores, twice as many hospitals, 50 fewer schools, and 5 more police stations. The total number of buildings required for this project is 2175. How many stores does the reference city have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?<|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
The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?<|im_end|>
<|im_start|>assistant
<Z_176><Z_116><Z_270><Z_484><Z_404><Z_84><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?<|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
The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?<|im_end|>
<|im_start|>assistant
<Z_76><Z_51><Z_407><Z_113><Z_41><Z_53><ANSWER>

Question: The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. He initially had 80 white cookies. How many cookies are remaining within the cookie jar altogether?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. He initially had 80 white cookies. How many cookies are remaining within the cookie jar altogether?<|im_end|>
<|im_start|>assistant
<Z_5><Z_304><Z_475><Z_431><Z_77><Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. He initially had 80 white cookies. How many cookies are remaining within the cookie jar altogether?<|im_end|>
<|im_start|>assistant
<Z_273><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. He initially had 80 white cookies. How many cookies are remaining within the cookie jar altogether?<|im_end|>
<|im_start|>assistant
<Z_155><Z_395><Z_274><Z_154><Z_148><Z_75><ANSWER>

Question: Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. He initially had 80 white cookies. How many cookies are remaining within the cookie jar altogether?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lucy has an aquarium with some fish. Lucy wants to buy 68 more fish. Then Lucy would have 280 fish. How many fish did Lucy have initially?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lucy has an aquarium with some fish. Lucy wants to buy 68 more fish. Then Lucy would have 280 fish. How many fish did Lucy have initially?<|im_end|>
<|im_start|>assistant
<Z_195><Z_409><Z_383><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lucy has an aquarium with some fish. Lucy wants to buy 68 more fish. Then Lucy would have 280 fish. How many fish did Lucy have initially?<|im_end|>
<|im_start|>assistant
<Z_246><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lucy has an aquarium with some fish. Lucy wants to buy 68 more fish. Then Lucy would have 280 fish. How many fish did Lucy have initially?<|im_end|>
<|im_start|>assistant
<Z_112><Z_508><Z_266><Z_296><ANSWER>

Question: Lucy has an aquarium with some fish. Lucy wants to buy 68 more fish. Then Lucy would have 280 fish. How many fish did Lucy have initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the coefficient of $x^2$ in the expansion of $(1-x)^4(1-)^3$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the coefficient of $x^2$ in the expansion of $(1-x)^4(1-)^3$.<|im_end|>
<|im_start|>assistant
<Z_358><Z_361><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the coefficient of $x^2$ in the expansion of $(1-x)^4(1-)^3$.<|im_end|>
<|im_start|>assistant
<Z_184><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the coefficient of $x^2$ in the expansion of $(1-x)^4(1-)^3$.<|im_end|>
<|im_start|>assistant
<Z_241><Z_373><Z_85><ANSWER>

Question: Determine the coefficient of $x^2$ in the expansion of $(1-x)^4(1-)^3$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p$ and $q$ are polynomials such that $i(x) = p(q(x))^2 - q(x)^3$. Find the degree of $q(x)$ given that the degree of $i(x)$ is $12$ and the degree of $p(x)$ is $4$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p$ and $q$ are polynomials such that $i(x) = p(q(x))^2 - q(x)^3$. Find the degree of $q(x)$ given that the degree of $i(x)$ is $12$ and the degree of $p(x)$ is $4$.<|im_end|>
<|im_start|>assistant
<Z_72><Z_40><Z_372><Z_418><Z_98><Z_252><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p$ and $q$ are polynomials such that $i(x) = p(q(x))^2 - q(x)^3$. Find the degree of $q(x)$ given that the degree of $i(x)$ is $12$ and the degree of $p(x)$ is $4$.<|im_end|>
<|im_start|>assistant
<Z_189><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p$ and $q$ are polynomials such that $i(x) = p(q(x))^2 - q(x)^3$. Find the degree of $q(x)$ given that the degree of $i(x)$ is $12$ and the degree of $p(x)$ is $4$.<|im_end|>
<|im_start|>assistant
<Z_395><Z_308><Z_381><Z_352><Z_108><Z_511><ANSWER>

Question: Suppose $p$ and $q$ are polynomials such that $i(x) = p(q(x))^2 - q(x)^3$. Find the degree of $q(x)$ given that the degree of $i(x)$ is $12$ and the degree of $p(x)$ is $4$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mr. Banks and Ms. Elizabeth are entrepreneurs who invest in different income streams and earn revenue from the investments. Mr. Banks received $500 in revenue from each of his 8 investments, while Ms. Elizabeth received $900 from each of her investment streams. Ms. Elizabeth made $500 more from her investments than Mr. Banks. How many investment streams did Ms. Elizabeth have?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mr. Banks and Ms. Elizabeth are entrepreneurs who invest in different income streams and earn revenue from the investments. Mr. Banks received $500 in revenue from each of his 8 investments, while Ms. Elizabeth received $900 from each of her investment streams. Ms. Elizabeth made $500 more from her investments than Mr. Banks. How many investment streams did Ms. Elizabeth have?<|im_end|>
<|im_start|>assistant
<Z_5><Z_499><Z_118><Z_54><Z_380><Z_305><Z_483><Z_48><Z_465><Z_488><Z_392><Z_376><Z_407><Z_193><Z_354><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mr. Banks and Ms. Elizabeth are entrepreneurs who invest in different income streams and earn revenue from the investments. Mr. Banks received $500 in revenue from each of his 8 investments, while Ms. Elizabeth received $900 from each of her investment streams. Ms. Elizabeth made $500 more from her investments than Mr. Banks. How many investment streams did Ms. Elizabeth have?<|im_end|>
<|im_start|>assistant
<Z_6><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mr. Banks and Ms. Elizabeth are entrepreneurs who invest in different income streams and earn revenue from the investments. Mr. Banks received $500 in revenue from each of his 8 investments, while Ms. Elizabeth received $900 from each of her investment streams. Ms. Elizabeth made $500 more from her investments than Mr. Banks. How many investment streams did Ms. Elizabeth have?<|im_end|>
<|im_start|>assistant
<Z_409><Z_167><Z_75><Z_204><Z_26><Z_302><Z_395><Z_202><Z_35><Z_11><Z_109><Z_133><Z_88><Z_256><Z_371><ANSWER>

Question: Mr. Banks and Ms. Elizabeth are entrepreneurs who invest in different income streams and earn revenue from the investments. Mr. Banks received $500 in revenue from each of his 8 investments, while Ms. Elizabeth received $900 from each of her investment streams. Ms. Elizabeth made $500 more from her investments than Mr. Banks. How many investment streams did Ms. Elizabeth have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bob has to hire someone to fix his garden. A storm destroyed some of his rose bushes. He decides to replant all the rose bushes. Each rose bush costs $150. He also needs to pay a gardener $30 an hour, 5 hours each day for 4 days. The final expense is 100 cubic feet of soil sold for $5 per cubic foot. The entire gardening project cost $4100. How many rose bushes were destroyed?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bob has to hire someone to fix his garden. A storm destroyed some of his rose bushes. He decides to replant all the rose bushes. Each rose bush costs $150. He also needs to pay a gardener $30 an hour, 5 hours each day for 4 days. The final expense is 100 cubic feet of soil sold for $5 per cubic foot. The entire gardening project cost $4100. How many rose bushes were destroyed?<|im_end|>
<|im_start|>assistant
<Z_358><Z_149><Z_77><Z_510><Z_319><Z_180><Z_339><Z_506><Z_315><Z_280><Z_172><Z_221><Z_510><Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bob has to hire someone to fix his garden. A storm destroyed some of his rose bushes. He decides to replant all the rose bushes. Each rose bush costs $150. He also needs to pay a gardener $30 an hour, 5 hours each day for 4 days. The final expense is 100 cubic feet of soil sold for $5 per cubic foot. The entire gardening project cost $4100. How many rose bushes were destroyed?<|im_end|>
<|im_start|>assistant
<Z_309><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bob has to hire someone to fix his garden. A storm destroyed some of his rose bushes. He decides to replant all the rose bushes. Each rose bush costs $150. He also needs to pay a gardener $30 an hour, 5 hours each day for 4 days. The final expense is 100 cubic feet of soil sold for $5 per cubic foot. The entire gardening project cost $4100. How many rose bushes were destroyed?<|im_end|>
<|im_start|>assistant
<Z_296><Z_204><Z_432><Z_249><Z_50><Z_176><Z_193><Z_146><Z_382><Z_500><Z_51><Z_357><Z_16><Z_511><ANSWER>

Question: Bob has to hire someone to fix his garden. A storm destroyed some of his rose bushes. He decides to replant all the rose bushes. Each rose bush costs $150. He also needs to pay a gardener $30 an hour, 5 hours each day for 4 days. The final expense is 100 cubic feet of soil sold for $5 per cubic foot. The entire gardening project cost $4100. How many rose bushes were destroyed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tamia is preparing a grand feast. She decides to use a variety of vegetables for her meal, including bell peppers, onions, and zucchinis. She has 10 bell peppers, 7 onions and 15 zucchinis.

First, Tamia prepares the bell peppers. She thinly slices 25% of the bell peppers into 20 pieces each. The rest of the bell peppers she cuts into 10 large pieces each. Then, she takes half of these large pieces and further cuts them into 3 smaller portions each.

Next, she moves on to the onions. She slices 50% of the onions into 18 thin slices each. But for the other onions, she prefers larger chunks and cuts these into 8 pieces each.

Finally, Tamia works with the zucchinis. From her total batch, 30% are cut into 15 slices each. The remaining ones are cut into 8 pieces each.

In completing this complex task, how many slices and pieces of vegetables in total does Tamia have? Note: A piece can be a slice or a chunk.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tamia is preparing a grand feast. She decides to use a variety of vegetables for her meal, including bell peppers, onions, and zucchinis. She has 10 bell peppers, 7 onions and 15 zucchinis.

First, Tamia prepares the bell peppers. She thinly slices 25% of the bell peppers into 20 pieces each. The rest of the bell peppers she cuts into 10 large pieces each. Then, she takes half of these large pieces and further cuts them into 3 smaller portions each.

Next, she moves on to the onions. She slices 50% of the onions into 18 thin slices each. But for the other onions, she prefers larger chunks and cuts these into 8 pieces each.

Finally, Tamia works with the zucchinis. From her total batch, 30% are cut into 15 slices each. The remaining ones are cut into 8 pieces each.

In completing this complex task, how many slices and pieces of vegetables in total does Tamia have? Note: A piece can be a slice or a chunk.<|im_end|>
<|im_start|>assistant
<Z_358><Z_223><Z_96><Z_340><Z_464><Z_292><Z_131><Z_447><Z_250><Z_213><Z_65><Z_373><Z_270><Z_315><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tamia is preparing a grand feast. She decides to use a variety of vegetables for her meal, including bell peppers, onions, and zucchinis. She has 10 bell peppers, 7 onions and 15 zucchinis.

First, Tamia prepares the bell peppers. She thinly slices 25% of the bell peppers into 20 pieces each. The rest of the bell peppers she cuts into 10 large pieces each. Then, she takes half of these large pieces and further cuts them into 3 smaller portions each.

Next, she moves on to the onions. She slices 50% of the onions into 18 thin slices each. But for the other onions, she prefers larger chunks and cuts these into 8 pieces each.

Finally, Tamia works with the zucchinis. From her total batch, 30% are cut into 15 slices each. The remaining ones are cut into 8 pieces each.

In completing this complex task, how many slices and pieces of vegetables in total does Tamia have? Note: A piece can be a slice or a chunk.<|im_end|>
<|im_start|>assistant
<Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tamia is preparing a grand feast. She decides to use a variety of vegetables for her meal, including bell peppers, onions, and zucchinis. She has 10 bell peppers, 7 onions and 15 zucchinis.

First, Tamia prepares the bell peppers. She thinly slices 25% of the bell peppers into 20 pieces each. The rest of the bell peppers she cuts into 10 large pieces each. Then, she takes half of these large pieces and further cuts them into 3 smaller portions each.

Next, she moves on to the onions. She slices 50% of the onions into 18 thin slices each. But for the other onions, she prefers larger chunks and cuts these into 8 pieces each.

Finally, Tamia works with the zucchinis. From her total batch, 30% are cut into 15 slices each. The remaining ones are cut into 8 pieces each.

In completing this complex task, how many slices and pieces of vegetables in total does Tamia have? Note: A piece can be a slice or a chunk.<|im_end|>
<|im_start|>assistant
<Z_68><Z_356><Z_223><Z_80><Z_268><Z_29><Z_279><Z_147><Z_270><Z_194><Z_279><Z_346><Z_397><Z_259><ANSWER>

Question:
Tamia is preparing a grand feast. She decides to use a variety of vegetables for her meal, including bell peppers, onions, and zucchinis. She has 10 bell peppers, 7 onions and 15 zucchinis.

First, Tamia prepares the bell peppers. She thinly slices 25% of the bell peppers into 20 pieces each. The rest of the bell peppers she cuts into 10 large pieces each. Then, she takes half of these large pieces and further cuts them into 3 smaller portions each.

Next, she moves on to the onions. She slices 50% of the onions into 18 thin slices each. But for the other onions, she prefers larger chunks and cuts these into 8 pieces each.

Finally, Tamia works with the zucchinis. From her total batch, 30% are cut into 15 slices each. The remaining ones are cut into 8 pieces each.

In completing this complex task, how many slices and pieces of vegetables in total does Tamia have? Note: A piece can be a slice or a chunk.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors. Given $\overrightarrow{AB} = 2\overrightarrow{a} + m\overrightarrow{b}$ and $\overrightarrow{CB} = \overrightarrow{a} + 3\overrightarrow{b}$. If points A, B, and C are collinear, then the value of $m$ is: ______.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors. Given $\overrightarrow{AB} = 2\overrightarrow{a} + m\overrightarrow{b}$ and $\overrightarrow{CB} = \overrightarrow{a} + 3\overrightarrow{b}$. If points A, B, and C are collinear, then the value of $m$ is: ______.<|im_end|>
<|im_start|>assistant
<Z_176><Z_273><Z_467><Z_300><Z_341><Z_256><Z_323><Z_286><Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors. Given $\overrightarrow{AB} = 2\overrightarrow{a} + m\overrightarrow{b}$ and $\overrightarrow{CB} = \overrightarrow{a} + 3\overrightarrow{b}$. If points A, B, and C are collinear, then the value of $m$ is: ______.<|im_end|>
<|im_start|>assistant
<Z_380><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors. Given $\overrightarrow{AB} = 2\overrightarrow{a} + m\overrightarrow{b}$ and $\overrightarrow{CB} = \overrightarrow{a} + 3\overrightarrow{b}$. If points A, B, and C are collinear, then the value of $m$ is: ______.<|im_end|>
<|im_start|>assistant
<Z_208><Z_410><Z_326><Z_452><Z_501><Z_0><Z_204><Z_345><Z_166><ANSWER>

Question: Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors. Given $\overrightarrow{AB} = 2\overrightarrow{a} + m\overrightarrow{b}$ and $\overrightarrow{CB} = \overrightarrow{a} + 3\overrightarrow{b}$. If points A, B, and C are collinear, then the value of $m$ is: ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $45 \times 55 + 45 \times 45$.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $45 \times 55 + 45 \times 45$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_214><Z_253><Z_372><Z_337><Z_188><Z_146><Z_276><Z_322><Z_147><Z_510><Z_323><Z_267><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $45 \times 55 + 45 \times 45$.<|im_end|>
<|im_start|>assistant
<Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $45 \times 55 + 45 \times 45$.<|im_end|>
<|im_start|>assistant
<Z_200><Z_107><Z_86><Z_74><Z_292><Z_290><Z_0><Z_271><Z_321><Z_481><Z_479><Z_235><Z_202><ANSWER>

Question: Compute $45 \times 55 + 45 \times 45$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book has 5 chapters. The first chapter is 60 pages long, the second chapter is 75 pages long, the third chapter is 56 pages long, and the fourth chapter is 42 pages long. If there are a total of 325 pages in the book, how many pages are in the fifth chapter, and what is the average page length per chapter?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book has 5 chapters. The first chapter is 60 pages long, the second chapter is 75 pages long, the third chapter is 56 pages long, and the fourth chapter is 42 pages long. If there are a total of 325 pages in the book, how many pages are in the fifth chapter, and what is the average page length per chapter?<|im_end|>
<|im_start|>assistant
<Z_5><Z_387><Z_442><Z_303><Z_475><Z_183><Z_413><Z_183><Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book has 5 chapters. The first chapter is 60 pages long, the second chapter is 75 pages long, the third chapter is 56 pages long, and the fourth chapter is 42 pages long. If there are a total of 325 pages in the book, how many pages are in the fifth chapter, and what is the average page length per chapter?<|im_end|>
<|im_start|>assistant
<Z_407><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book has 5 chapters. The first chapter is 60 pages long, the second chapter is 75 pages long, the third chapter is 56 pages long, and the fourth chapter is 42 pages long. If there are a total of 325 pages in the book, how many pages are in the fifth chapter, and what is the average page length per chapter?<|im_end|>
<|im_start|>assistant
<Z_374><Z_446><Z_394><Z_154><Z_71><Z_388><Z_194><Z_417><Z_449><ANSWER>

Question: A book has 5 chapters. The first chapter is 60 pages long, the second chapter is 75 pages long, the third chapter is 56 pages long, and the fourth chapter is 42 pages long. If there are a total of 325 pages in the book, how many pages are in the fifth chapter, and what is the average page length per chapter?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the end of a basketball tournament, each player shook hands once with every player from other teams (excluding themselves). Additionally, a coach shook hands with each player of his own team. If there were a total of 465 handshakes, what is the maximum number of handshakes that the coach could have participated in?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the end of a basketball tournament, each player shook hands once with every player from other teams (excluding themselves). Additionally, a coach shook hands with each player of his own team. If there were a total of 465 handshakes, what is the maximum number of handshakes that the coach could have participated in?<|im_end|>
<|im_start|>assistant
<Z_195><Z_185><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the end of a basketball tournament, each player shook hands once with every player from other teams (excluding themselves). Additionally, a coach shook hands with each player of his own team. If there were a total of 465 handshakes, what is the maximum number of handshakes that the coach could have participated in?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the end of a basketball tournament, each player shook hands once with every player from other teams (excluding themselves). Additionally, a coach shook hands with each player of his own team. If there were a total of 465 handshakes, what is the maximum number of handshakes that the coach could have participated in?<|im_end|>
<|im_start|>assistant
<Z_158><Z_421><ANSWER>

Question: At the end of a basketball tournament, each player shook hands once with every player from other teams (excluding themselves). Additionally, a coach shook hands with each player of his own team. If there were a total of 465 handshakes, what is the maximum number of handshakes that the coach could have participated in?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person is chosing between 2 cellular phone plans . One has a monthly fee of $ 50 for 500 minutes plus $ 0.35 for each minute over the 500 . The second plan has a monthly fee of $ 75 for 1000 minutes plus $ 0.45 for each minute over the 1000 . After how many minutes used are the monthly costs of the plans equal ?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person is chosing between 2 cellular phone plans . One has a monthly fee of $ 50 for 500 minutes plus $ 0.35 for each minute over the 500 . The second plan has a monthly fee of $ 75 for 1000 minutes plus $ 0.45 for each minute over the 1000 . After how many minutes used are the monthly costs of the plans equal ?<|im_end|>
<|im_start|>assistant
<Z_195><Z_345><Z_354><Z_281><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person is chosing between 2 cellular phone plans . One has a monthly fee of $ 50 for 500 minutes plus $ 0.35 for each minute over the 500 . The second plan has a monthly fee of $ 75 for 1000 minutes plus $ 0.45 for each minute over the 1000 . After how many minutes used are the monthly costs of the plans equal ?<|im_end|>
<|im_start|>assistant
<Z_170><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person is chosing between 2 cellular phone plans . One has a monthly fee of $ 50 for 500 minutes plus $ 0.35 for each minute over the 500 . The second plan has a monthly fee of $ 75 for 1000 minutes plus $ 0.45 for each minute over the 1000 . After how many minutes used are the monthly costs of the plans equal ?<|im_end|>
<|im_start|>assistant
<Z_132><Z_153><Z_321><Z_198><ANSWER>

Question: A person is chosing between 2 cellular phone plans . One has a monthly fee of $ 50 for 500 minutes plus $ 0.35 for each minute over the 500 . The second plan has a monthly fee of $ 75 for 1000 minutes plus $ 0.45 for each minute over the 1000 . After how many minutes used are the monthly costs of the plans equal ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $X+Y$ (in base 10), given the following addition problem in base 7:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
  & & & 5 & X & Y_{7}\\
+ & & & & 3 & 2_{7}\\
\cline{2-6}
  & & & 6 & 2 & X_{7}\\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $X+Y$ (in base 10), given the following addition problem in base 7:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
  & & & 5 & X & Y_{7}\\
+ & & & & 3 & 2_{7}\\
\cline{2-6}
  & & & 6 & 2 & X_{7}\\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_5><Z_289><Z_366><Z_416><Z_237><Z_265><Z_351><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $X+Y$ (in base 10), given the following addition problem in base 7:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
  & & & 5 & X & Y_{7}\\
+ & & & & 3 & 2_{7}\\
\cline{2-6}
  & & & 6 & 2 & X_{7}\\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_507><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $X+Y$ (in base 10), given the following addition problem in base 7:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
  & & & 5 & X & Y_{7}\\
+ & & & & 3 & 2_{7}\\
\cline{2-6}
  & & & 6 & 2 & X_{7}\\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_438><Z_160><Z_96><Z_45><Z_350><Z_129><Z_217><Z_86><ANSWER>

Question:
Find $X+Y$ (in base 10), given the following addition problem in base 7:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
  & & & 5 & X & Y_{7}\\
+ & & & & 3 & 2_{7}\\
\cline{2-6}
  & & & 6 & 2 & X_{7}\\
\end{array}
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider sets $\{6, 7, 8, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider sets $\{6, 7, 8, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?<|im_end|>
<|im_start|>assistant
<Z_176><Z_504><Z_155><Z_267><Z_469><Z_432><Z_510><Z_168><Z_235><Z_386><Z_105><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider sets $\{6, 7, 8, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider sets $\{6, 7, 8, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?<|im_end|>
<|im_start|>assistant
<Z_275><Z_112><Z_18><Z_35><Z_251><Z_64><Z_33><Z_348><Z_43><Z_420><Z_463><Z_81><ANSWER>

Question: Consider sets $\{6, 7, 8, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two diesel trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively . in what time will they be clear of each other from the moment they meet ?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two diesel trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively . in what time will they be clear of each other from the moment they meet ?<|im_end|>
<|im_start|>assistant
<Z_195><Z_461><Z_131><Z_156><Z_118><Z_287><Z_162><Z_495><Z_172><Z_257><Z_320><Z_483><Z_506><Z_189><Z_445><Z_342><Z_300><Z_281><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two diesel trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively . in what time will they be clear of each other from the moment they meet ?<|im_end|>
<|im_start|>assistant
<Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two diesel trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively . in what time will they be clear of each other from the moment they meet ?<|im_end|>
<|im_start|>assistant
<Z_145><Z_504><Z_154><Z_9><Z_508><Z_243><Z_475><Z_66><Z_473><Z_90><Z_392><Z_454><Z_53><Z_278><Z_414><Z_325><Z_116><Z_456><ANSWER>

Question: two diesel trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively . in what time will they be clear of each other from the moment they meet ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + 2\mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + 2\mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$<|im_end|>
<|im_start|>assistant
<Z_66><Z_133><Z_265><Z_280><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + 2\mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$<|im_end|>
<|im_start|>assistant
<Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + 2\mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$<|im_end|>
<|im_start|>assistant
<Z_101><Z_30><Z_66><Z_99><ANSWER>

Question: If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + 2\mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John climbs up 9 flights of stairs. Each flight is 10 feet. If each step is 18 inches, how many steps does he climb up?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John climbs up 9 flights of stairs. Each flight is 10 feet. If each step is 18 inches, how many steps does he climb up?<|im_end|>
<|im_start|>assistant
<Z_176><Z_131><Z_45><Z_284><Z_472><Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John climbs up 9 flights of stairs. Each flight is 10 feet. If each step is 18 inches, how many steps does he climb up?<|im_end|>
<|im_start|>assistant
<Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John climbs up 9 flights of stairs. Each flight is 10 feet. If each step is 18 inches, how many steps does he climb up?<|im_end|>
<|im_start|>assistant
<Z_259><Z_285><Z_89><Z_387><Z_392><Z_13><ANSWER>

Question: John climbs up 9 flights of stairs. Each flight is 10 feet. If each step is 18 inches, how many steps does he climb up?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_66><Z_99><Z_510><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_357><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_491><Z_15><Z_12><ANSWER>

Question:
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_66><Z_281><Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_222><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_172><Z_355><Z_34><ANSWER>

Question:
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|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
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|im_end|>
<|im_start|>assistant
<Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|im_end|>
<|im_start|>assistant
<Z_40><ANSWER>

Question: Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_5><Z_265><Z_380><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_332><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_211><Z_114><Z_23><ANSWER>

Question: Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_195><Z_299><Z_400><Z_233><Z_376><Z_417><Z_301><Z_187><Z_407><Z_125><Z_92><Z_256><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_481><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_247><Z_98><Z_384><Z_455><Z_313><Z_407><Z_347><Z_359><Z_453><Z_11><Z_141><Z_434><ANSWER>

Question: Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?