<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man , a woman and a boy can together complete a piece of work in 3 days . if a man alone can do it in 6 days and a boy alone in 12 days , how long will a woman take to complete the work ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man , a woman and a boy can together complete a piece of work in 3 days . if a man alone can do it in 6 days and a boy alone in 12 days , how long will a woman take to complete the work ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_246><Z_425><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man , a woman and a boy can together complete a piece of work in 3 days . if a man alone can do it in 6 days and a boy alone in 12 days , how long will a woman take to complete the work ?<|im_end|>
<|im_start|>assistant
<Z_233><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man , a woman and a boy can together complete a piece of work in 3 days . if a man alone can do it in 6 days and a boy alone in 12 days , how long will a woman take to complete the work ?<|im_end|>
<|im_start|>assistant
<Z_440><Z_467><Z_438><Z_476><ANSWER>

Question: a man , a woman and a boy can together complete a piece of work in 3 days . if a man alone can do it in 6 days and a boy alone in 12 days , how long will a woman take to complete the work ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that circle O is tangent to circle C: x²+y²-6y+8=0 at point M(0,2), and passes through point N(2,0).
(1) Find the equation of circle O;
(2) If line L: y=kx-(k+1) intercepts two arc lengths on circle O with a ratio of 3:1, find the value of the real number k.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that circle O is tangent to circle C: x²+y²-6y+8=0 at point M(0,2), and passes through point N(2,0).
(1) Find the equation of circle O;
(2) If line L: y=kx-(k+1) intercepts two arc lengths on circle O with a ratio of 3:1, find the value of the real number k.<|im_end|>
<|im_start|>assistant
<Z_477><Z_244><Z_461><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that circle O is tangent to circle C: x²+y²-6y+8=0 at point M(0,2), and passes through point N(2,0).
(1) Find the equation of circle O;
(2) If line L: y=kx-(k+1) intercepts two arc lengths on circle O with a ratio of 3:1, find the value of the real number k.<|im_end|>
<|im_start|>assistant
<Z_412><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that circle O is tangent to circle C: x²+y²-6y+8=0 at point M(0,2), and passes through point N(2,0).
(1) Find the equation of circle O;
(2) If line L: y=kx-(k+1) intercepts two arc lengths on circle O with a ratio of 3:1, find the value of the real number k.<|im_end|>
<|im_start|>assistant
<Z_205><Z_482><Z_459><ANSWER>

Question:
Given that circle O is tangent to circle C: x²+y²-6y+8=0 at point M(0,2), and passes through point N(2,0).
(1) Find the equation of circle O;
(2) If line L: y=kx-(k+1) intercepts two arc lengths on circle O with a ratio of 3:1, find the value of the real number k.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers \( n \) are there with the property that the product of the digits of \( n \) is 0, where \( 5000 \leq n \leq 6000 \)?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers \( n \) are there with the property that the product of the digits of \( n \) is 0, where \( 5000 \leq n \leq 6000 \)?<|im_end|>
<|im_start|>assistant
<Z_369><Z_219><Z_283><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers \( n \) are there with the property that the product of the digits of \( n \) is 0, where \( 5000 \leq n \leq 6000 \)?<|im_end|>
<|im_start|>assistant
<Z_174><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers \( n \) are there with the property that the product of the digits of \( n \) is 0, where \( 5000 \leq n \leq 6000 \)?<|im_end|>
<|im_start|>assistant
<Z_253><Z_241><Z_502><ANSWER>

Question: How many integers \( n \) are there with the property that the product of the digits of \( n \) is 0, where \( 5000 \leq n \leq 6000 \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
sandy walked 20 meters towards south . then sandy turned to her left and walked 20 meters . she then turned to her left and walked 20 meters . she then turned to her right and walked 20 meters . what distance is she from the starting point and in which direction ?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
sandy walked 20 meters towards south . then sandy turned to her left and walked 20 meters . she then turned to her left and walked 20 meters . she then turned to her right and walked 20 meters . what distance is she from the starting point and in which direction ?<|im_end|>
<|im_start|>assistant
<Z_369><Z_488><Z_135><Z_123><Z_461><Z_179><Z_263><Z_255><Z_295><Z_255><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
sandy walked 20 meters towards south . then sandy turned to her left and walked 20 meters . she then turned to her left and walked 20 meters . she then turned to her right and walked 20 meters . what distance is she from the starting point and in which direction ?<|im_end|>
<|im_start|>assistant
<Z_320><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
sandy walked 20 meters towards south . then sandy turned to her left and walked 20 meters . she then turned to her left and walked 20 meters . she then turned to her right and walked 20 meters . what distance is she from the starting point and in which direction ?<|im_end|>
<|im_start|>assistant
<Z_2><Z_26><Z_107><Z_11><Z_297><Z_464><Z_53><Z_222><Z_422><Z_329><Z_246><ANSWER>

Question: sandy walked 20 meters towards south . then sandy turned to her left and walked 20 meters . she then turned to her left and walked 20 meters . she then turned to her right and walked 20 meters . what distance is she from the starting point and in which direction ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( O \) and \( H \) be the circumcenter and orthocenter of triangle \( ABC \), respectively. Let \( a, b, \) and \( c \) denote the side lengths, and let \( R \) denote the circumradius. Find \( OH^2 \) if \( R = 5 \) and \( a^2 + b^2 + c^2 = 50 \).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( O \) and \( H \) be the circumcenter and orthocenter of triangle \( ABC \), respectively. Let \( a, b, \) and \( c \) denote the side lengths, and let \( R \) denote the circumradius. Find \( OH^2 \) if \( R = 5 \) and \( a^2 + b^2 + c^2 = 50 \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_98><Z_297><Z_507><Z_111><Z_256><Z_432><Z_344><Z_149><Z_269><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( O \) and \( H \) be the circumcenter and orthocenter of triangle \( ABC \), respectively. Let \( a, b, \) and \( c \) denote the side lengths, and let \( R \) denote the circumradius. Find \( OH^2 \) if \( R = 5 \) and \( a^2 + b^2 + c^2 = 50 \).<|im_end|>
<|im_start|>assistant
<Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( O \) and \( H \) be the circumcenter and orthocenter of triangle \( ABC \), respectively. Let \( a, b, \) and \( c \) denote the side lengths, and let \( R \) denote the circumradius. Find \( OH^2 \) if \( R = 5 \) and \( a^2 + b^2 + c^2 = 50 \).<|im_end|>
<|im_start|>assistant
<Z_419><Z_32><Z_172><Z_445><Z_376><Z_363><Z_343><Z_277><Z_462><Z_102><Z_492><ANSWER>

Question: Let \( O \) and \( H \) be the circumcenter and orthocenter of triangle \( ABC \), respectively. Let \( a, b, \) and \( c \) denote the side lengths, and let \( R \) denote the circumradius. Find \( OH^2 \) if \( R = 5 \) and \( a^2 + b^2 + c^2 = 50 \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of the solutions for the equation $x^2 - 6x - 22 = 4x + 24$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of the solutions for the equation $x^2 - 6x - 22 = 4x + 24$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of the solutions for the equation $x^2 - 6x - 22 = 4x + 24$.<|im_end|>
<|im_start|>assistant
<Z_256><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of the solutions for the equation $x^2 - 6x - 22 = 4x + 24$.<|im_end|>
<|im_start|>assistant
<Z_441><Z_495><ANSWER>

Question: Determine the sum of the solutions for the equation $x^2 - 6x - 22 = 4x + 24$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
David obtained 70 marks in English, 60 in Mathematics, 78 in Physics, 60 in Chemistry, and some marks in Biology. His average marks are 66.6. What are his marks in Biology?<|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
David obtained 70 marks in English, 60 in Mathematics, 78 in Physics, 60 in Chemistry, and some marks in Biology. His average marks are 66.6. What are his marks in Biology?<|im_end|>
<|im_start|>assistant
<Z_24><Z_432><Z_296><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
David obtained 70 marks in English, 60 in Mathematics, 78 in Physics, 60 in Chemistry, and some marks in Biology. His average marks are 66.6. What are his marks in Biology?<|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
David obtained 70 marks in English, 60 in Mathematics, 78 in Physics, 60 in Chemistry, and some marks in Biology. His average marks are 66.6. What are his marks in Biology?<|im_end|>
<|im_start|>assistant
<Z_243><Z_473><Z_211><ANSWER>

Question: David obtained 70 marks in English, 60 in Mathematics, 78 in Physics, 60 in Chemistry, and some marks in Biology. His average marks are 66.6. What are his marks in Biology?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of  $a, b, c,$  and  $d,$  the equation  $x^4+ax^3+bx^2+cx+d=0$  has four non-real roots.  The product of two of these roots is  $13+i$  and the sum of the other two roots is  $3+4i,$  where  $i=\sqrt{-1}.$   Find  $b.$ <|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of  $a, b, c,$  and  $d,$  the equation  $x^4+ax^3+bx^2+cx+d=0$  has four non-real roots.  The product of two of these roots is  $13+i$  and the sum of the other two roots is  $3+4i,$  where  $i=\sqrt{-1}.$   Find  $b.$ <|im_end|>
<|im_start|>assistant
<Z_369><Z_197><Z_497><Z_344><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of  $a, b, c,$  and  $d,$  the equation  $x^4+ax^3+bx^2+cx+d=0$  has four non-real roots.  The product of two of these roots is  $13+i$  and the sum of the other two roots is  $3+4i,$  where  $i=\sqrt{-1}.$   Find  $b.$ <|im_end|>
<|im_start|>assistant
<Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of  $a, b, c,$  and  $d,$  the equation  $x^4+ax^3+bx^2+cx+d=0$  has four non-real roots.  The product of two of these roots is  $13+i$  and the sum of the other two roots is  $3+4i,$  where  $i=\sqrt{-1}.$   Find  $b.$ <|im_end|>
<|im_start|>assistant
<Z_461><Z_459><Z_278><Z_434><ANSWER>

Question: For certain real values of  $a, b, c,$  and  $d,$  the equation  $x^4+ax^3+bx^2+cx+d=0$  has four non-real roots.  The product of two of these roots is  $13+i$  and the sum of the other two roots is  $3+4i,$  where  $i=\sqrt{-1}.$   Find  $b.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal?<|im_end|>
<|im_start|>assistant
<Z_24><Z_377><Z_164><Z_144><Z_240><Z_206><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal?<|im_end|>
<|im_start|>assistant
<Z_198><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal?<|im_end|>
<|im_start|>assistant
<Z_432><Z_444><Z_268><Z_278><Z_13><Z_397><ANSWER>

Question: What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.<|im_end|>
<|im_start|>assistant
<Z_369><Z_359><Z_302><Z_335><Z_503><Z_130><Z_423><Z_59><Z_446><Z_267><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.<|im_end|>
<|im_start|>assistant
<Z_7><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.<|im_end|>
<|im_start|>assistant
<Z_335><Z_235><Z_488><Z_252><Z_65><Z_84><Z_197><Z_163><Z_0><Z_253><Z_358><ANSWER>

Question: In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The product of the digits of the positive integer \( N \) is 20. One of the following could not be the product of the digits of \( N+1 \). Which is it?
A 24
B 25
C 30
D 35
E 40<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The product of the digits of the positive integer \( N \) is 20. One of the following could not be the product of the digits of \( N+1 \). Which is it?
A 24
B 25
C 30
D 35
E 40<|im_end|>
<|im_start|>assistant
<Z_369><Z_468><Z_378><Z_64><Z_185><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The product of the digits of the positive integer \( N \) is 20. One of the following could not be the product of the digits of \( N+1 \). Which is it?
A 24
B 25
C 30
D 35
E 40<|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
The product of the digits of the positive integer \( N \) is 20. One of the following could not be the product of the digits of \( N+1 \). Which is it?
A 24
B 25
C 30
D 35
E 40<|im_end|>
<|im_start|>assistant
<Z_339><Z_103><Z_416><Z_9><Z_320><ANSWER>

Question:
The product of the digits of the positive integer \( N \) is 20. One of the following could not be the product of the digits of \( N+1 \). Which is it?
A 24
B 25
C 30
D 35
E 40


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on a certain planet where people usually tend to live for more than a century , the ratio of present ages of father and son is 7 : 4 . 10 years later , the ratio of their ages will be 5 : 3 . what is the sum of their present ages ?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on a certain planet where people usually tend to live for more than a century , the ratio of present ages of father and son is 7 : 4 . 10 years later , the ratio of their ages will be 5 : 3 . what is the sum of their present ages ?<|im_end|>
<|im_start|>assistant
<Z_369><Z_465><Z_213><Z_274><Z_440><Z_420><Z_487><Z_426><Z_274><Z_487><Z_192><Z_449><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on a certain planet where people usually tend to live for more than a century , the ratio of present ages of father and son is 7 : 4 . 10 years later , the ratio of their ages will be 5 : 3 . what is the sum of their present ages ?<|im_end|>
<|im_start|>assistant
<Z_55><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on a certain planet where people usually tend to live for more than a century , the ratio of present ages of father and son is 7 : 4 . 10 years later , the ratio of their ages will be 5 : 3 . what is the sum of their present ages ?<|im_end|>
<|im_start|>assistant
<Z_448><Z_255><Z_18><Z_55><Z_383><Z_314><Z_109><Z_6><Z_308><Z_482><Z_140><Z_329><ANSWER>

Question: on a certain planet where people usually tend to live for more than a century , the ratio of present ages of father and son is 7 : 4 . 10 years later , the ratio of their ages will be 5 : 3 . what is the sum of their present ages ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(18)$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(18)$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_486><Z_322><Z_493><Z_266><Z_37><Z_295><Z_395><Z_310><Z_456><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(18)$.<|im_end|>
<|im_start|>assistant
<Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(18)$.<|im_end|>
<|im_start|>assistant
<Z_140><Z_202><Z_33><Z_184><Z_234><Z_147><Z_309><Z_7><Z_468><Z_86><Z_217><ANSWER>

Question:
Let $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(18)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two concentric circles have radii of 15 and 25 units, respectively. A shaded region exists between these two circles. Determine the radius of a fourth circle whose area is the same as that of the shaded region. Give your answer in simplest radical form.<|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
Two concentric circles have radii of 15 and 25 units, respectively. A shaded region exists between these two circles. Determine the radius of a fourth circle whose area is the same as that of the shaded region. Give your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_24><Z_53><Z_503><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two concentric circles have radii of 15 and 25 units, respectively. A shaded region exists between these two circles. Determine the radius of a fourth circle whose area is the same as that of the shaded region. Give your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_429><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two concentric circles have radii of 15 and 25 units, respectively. A shaded region exists between these two circles. Determine the radius of a fourth circle whose area is the same as that of the shaded region. Give your answer in simplest radical form.<|im_end|>
<|im_start|>assistant
<Z_434><Z_511><Z_306><ANSWER>

Question: Two concentric circles have radii of 15 and 25 units, respectively. A shaded region exists between these two circles. Determine the radius of a fourth circle whose area is the same as that of the shaded region. Give 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
In how many ways can 5 books be selected from a shelf of 8 books if the order in which the books are selected does not matter?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can 5 books be selected from a shelf of 8 books if the order in which the books are selected does not matter?<|im_end|>
<|im_start|>assistant
<Z_477><Z_319><Z_159><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can 5 books be selected from a shelf of 8 books if the order in which the books are selected does not matter?<|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
In how many ways can 5 books be selected from a shelf of 8 books if the order in which the books are selected does not matter?<|im_end|>
<|im_start|>assistant
<Z_182><Z_8><Z_368><ANSWER>

Question: In how many ways can 5 books be selected from a shelf of 8 books if the order in which the books are selected does not matter?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the positive whole number divisors of 360?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the positive whole number divisors of 360?<|im_end|>
<|im_start|>assistant
<Z_369><Z_255><Z_240><Z_393><Z_378><Z_422><Z_485><Z_151><Z_192><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the positive whole number divisors of 360?<|im_end|>
<|im_start|>assistant
<Z_486><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the positive whole number divisors of 360?<|im_end|>
<|im_start|>assistant
<Z_212><Z_437><Z_344><Z_466><Z_129><Z_158><Z_187><Z_420><Z_488><ANSWER>

Question: What is the sum of the positive whole number divisors of 360?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 6 minutes he is on a red dot? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
[/asy]<|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
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 6 minutes he is on a red dot? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_477><Z_210><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 6 minutes he is on a red dot? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_129><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 6 minutes he is on a red dot? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_145><Z_263><ANSWER>

Question:
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 6 minutes he is on a red dot? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?<|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
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?<|im_end|>
<|im_start|>assistant
<Z_369><Z_433><Z_105><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?<|im_end|>
<|im_start|>assistant
<Z_287><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?<|im_end|>
<|im_start|>assistant
<Z_155><Z_236><Z_273><ANSWER>

Question:
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
15 men can complete a work in 10 days while 20 boys can complete the same work in 15 days. How many days will 10 men and 10 boys together take to complete the same work?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
15 men can complete a work in 10 days while 20 boys can complete the same work in 15 days. How many days will 10 men and 10 boys together take to complete the same work?<|im_end|>
<|im_start|>assistant
<Z_369><Z_29><Z_195><Z_456><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
15 men can complete a work in 10 days while 20 boys can complete the same work in 15 days. How many days will 10 men and 10 boys together take to complete the same work?<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
15 men can complete a work in 10 days while 20 boys can complete the same work in 15 days. How many days will 10 men and 10 boys together take to complete the same work?<|im_end|>
<|im_start|>assistant
<Z_246><Z_345><Z_345><Z_408><ANSWER>

Question: 15 men can complete a work in 10 days while 20 boys can complete the same work in 15 days. How many days will 10 men and 10 boys together take to complete the same work?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The mass of a certain volume of a substance is some kg under certain conditions. The volume in cubic centimeters of 1 gram of this substance under these conditions is 2. What is the mass of 1 cubic meter of this substance under these conditions?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The mass of a certain volume of a substance is some kg under certain conditions. The volume in cubic centimeters of 1 gram of this substance under these conditions is 2. What is the mass of 1 cubic meter of this substance under these conditions?<|im_end|>
<|im_start|>assistant
<Z_369><Z_381><Z_427><Z_103><Z_232><Z_162><Z_106><Z_272><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The mass of a certain volume of a substance is some kg under certain conditions. The volume in cubic centimeters of 1 gram of this substance under these conditions is 2. What is the mass of 1 cubic meter of this substance under these conditions?<|im_end|>
<|im_start|>assistant
<Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The mass of a certain volume of a substance is some kg under certain conditions. The volume in cubic centimeters of 1 gram of this substance under these conditions is 2. What is the mass of 1 cubic meter of this substance under these conditions?<|im_end|>
<|im_start|>assistant
<Z_423><Z_507><Z_68><Z_122><Z_295><Z_44><Z_205><Z_35><ANSWER>

Question: The mass of a certain volume of a substance is some kg under certain conditions. The volume in cubic centimeters of 1 gram of this substance under these conditions is 2. What is the mass of 1 cubic meter of this substance under these conditions?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\overrightarrow{m}=(\cos \frac {x}{2},\sin \frac {x}{2})$ and $\overrightarrow{n}=(- \sqrt {3},1)$, find the maximum value of $|\overrightarrow{m}- \overrightarrow{n}|$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\overrightarrow{m}=(\cos \frac {x}{2},\sin \frac {x}{2})$ and $\overrightarrow{n}=(- \sqrt {3},1)$, find the maximum value of $|\overrightarrow{m}- \overrightarrow{n}|$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_287><Z_231><Z_456><Z_327><Z_214><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\overrightarrow{m}=(\cos \frac {x}{2},\sin \frac {x}{2})$ and $\overrightarrow{n}=(- \sqrt {3},1)$, find the maximum value of $|\overrightarrow{m}- \overrightarrow{n}|$.<|im_end|>
<|im_start|>assistant
<Z_203><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\overrightarrow{m}=(\cos \frac {x}{2},\sin \frac {x}{2})$ and $\overrightarrow{n}=(- \sqrt {3},1)$, find the maximum value of $|\overrightarrow{m}- \overrightarrow{n}|$.<|im_end|>
<|im_start|>assistant
<Z_457><Z_383><Z_358><Z_328><Z_315><Z_436><ANSWER>

Question: Given $\overrightarrow{m}=(\cos \frac {x}{2},\sin \frac {x}{2})$ and $\overrightarrow{n}=(- \sqrt {3},1)$, find the maximum value of $|\overrightarrow{m}- \overrightarrow{n}|$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.''  In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words).  How many valid 3-word sentences are there in Gnollish?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.''  In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words).  How many valid 3-word sentences are there in Gnollish?<|im_end|>
<|im_start|>assistant
<Z_477><Z_458><Z_446><Z_230><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.''  In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words).  How many valid 3-word sentences are there in Gnollish?<|im_end|>
<|im_start|>assistant
<Z_492><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.''  In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words).  How many valid 3-word sentences are there in Gnollish?<|im_end|>
<|im_start|>assistant
<Z_137><Z_411><Z_500><Z_357><ANSWER>

Question: The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.''  In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words).  How many valid 3-word sentences are there in Gnollish?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the grade point average of one third of the classroom is 30 ; the grade point average of the rest is 33 . what is the grade point average of the whole class ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the grade point average of one third of the classroom is 30 ; the grade point average of the rest is 33 . what is the grade point average of the whole class ?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the grade point average of one third of the classroom is 30 ; the grade point average of the rest is 33 . what is the grade point average of the whole class ?<|im_end|>
<|im_start|>assistant
<Z_132><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the grade point average of one third of the classroom is 30 ; the grade point average of the rest is 33 . what is the grade point average of the whole class ?<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

Question: the grade point average of one third of the classroom is 30 ; the grade point average of the rest is 33 . what is the grade point average of the whole class ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 4 men can colour a certain length of cloth in 2 days, then 8 men can colour 36 m long cloth in 0.75 days. What was the initial length of the cloth?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 4 men can colour a certain length of cloth in 2 days, then 8 men can colour 36 m long cloth in 0.75 days. What was the initial length of the cloth?<|im_end|>
<|im_start|>assistant
<Z_477><Z_403><Z_297><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 4 men can colour a certain length of cloth in 2 days, then 8 men can colour 36 m long cloth in 0.75 days. What was the initial length of the cloth?<|im_end|>
<|im_start|>assistant
<Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 4 men can colour a certain length of cloth in 2 days, then 8 men can colour 36 m long cloth in 0.75 days. What was the initial length of the cloth?<|im_end|>
<|im_start|>assistant
<Z_361><Z_10><Z_166><Z_204><ANSWER>

Question: If 4 men can colour a certain length of cloth in 2 days, then 8 men can colour 36 m long cloth in 0.75 days. What was the initial length of the cloth?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compare the combined area in square inches of the front and back of a trapezoidal sheet of paper with base lengths 11 inches and 19 inches and height of 10 inches, to that of another trapezoidal sheet of paper with base lengths 9.5 inches and 11 inches and height of 8 inches. How much greater is the combined area of the first trapezoidal sheet of paper than the second one?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compare the combined area in square inches of the front and back of a trapezoidal sheet of paper with base lengths 11 inches and 19 inches and height of 10 inches, to that of another trapezoidal sheet of paper with base lengths 9.5 inches and 11 inches and height of 8 inches. How much greater is the combined area of the first trapezoidal sheet of paper than the second one?<|im_end|>
<|im_start|>assistant
<Z_477><Z_227><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compare the combined area in square inches of the front and back of a trapezoidal sheet of paper with base lengths 11 inches and 19 inches and height of 10 inches, to that of another trapezoidal sheet of paper with base lengths 9.5 inches and 11 inches and height of 8 inches. How much greater is the combined area of the first trapezoidal sheet of paper than the second one?<|im_end|>
<|im_start|>assistant
<Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compare the combined area in square inches of the front and back of a trapezoidal sheet of paper with base lengths 11 inches and 19 inches and height of 10 inches, to that of another trapezoidal sheet of paper with base lengths 9.5 inches and 11 inches and height of 8 inches. How much greater is the combined area of the first trapezoidal sheet of paper than the second one?<|im_end|>
<|im_start|>assistant
<Z_365><Z_349><ANSWER>

Question: Compare the combined area in square inches of the front and back of a trapezoidal sheet of paper with base lengths 11 inches and 19 inches and height of 10 inches, to that of another trapezoidal sheet of paper with base lengths 9.5 inches and 11 inches and height of 8 inches. How much greater is the combined area of the first trapezoidal sheet of paper than the second one?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it is the new year and mandy has made a resolution to lose weight this year . she plans to exercise and do yoga . for exercise she plans to workout at the gym and ride her bicycle in the ratio of 2 : 3 everyday . she will also do yoga in the ratio , yoga : exercise = 2 : 3 . if she rides her bike for 12 minutes , how much time will she spend doing yoga ? ( rounded to minutes )<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it is the new year and mandy has made a resolution to lose weight this year . she plans to exercise and do yoga . for exercise she plans to workout at the gym and ride her bicycle in the ratio of 2 : 3 everyday . she will also do yoga in the ratio , yoga : exercise = 2 : 3 . if she rides her bike for 12 minutes , how much time will she spend doing yoga ? ( rounded to minutes )<|im_end|>
<|im_start|>assistant
<Z_369><Z_273><Z_496><Z_138><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it is the new year and mandy has made a resolution to lose weight this year . she plans to exercise and do yoga . for exercise she plans to workout at the gym and ride her bicycle in the ratio of 2 : 3 everyday . she will also do yoga in the ratio , yoga : exercise = 2 : 3 . if she rides her bike for 12 minutes , how much time will she spend doing yoga ? ( rounded to minutes )<|im_end|>
<|im_start|>assistant
<Z_479><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it is the new year and mandy has made a resolution to lose weight this year . she plans to exercise and do yoga . for exercise she plans to workout at the gym and ride her bicycle in the ratio of 2 : 3 everyday . she will also do yoga in the ratio , yoga : exercise = 2 : 3 . if she rides her bike for 12 minutes , how much time will she spend doing yoga ? ( rounded to minutes )<|im_end|>
<|im_start|>assistant
<Z_399><Z_436><Z_277><Z_471><ANSWER>

Question: it is the new year and mandy has made a resolution to lose weight this year . she plans to exercise and do yoga . for exercise she plans to workout at the gym and ride her bicycle in the ratio of 2 : 3 everyday . she will also do yoga in the ratio , yoga : exercise = 2 : 3 . if she rides her bike for 12 minutes , how much time will she spend doing yoga ? ( rounded to minutes )


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A waiter had some customers in his section. Seventeen of them left and the rest of his tables had nine people at each table. He had 5 tables. How many customers did he have initially in his section?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A waiter had some customers in his section. Seventeen of them left and the rest of his tables had nine people at each table. He had 5 tables. How many customers did he have initially in his section?<|im_end|>
<|im_start|>assistant
<Z_369><Z_30><Z_200><Z_45><Z_219><Z_327><Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A waiter had some customers in his section. Seventeen of them left and the rest of his tables had nine people at each table. He had 5 tables. How many customers did he have initially in his section?<|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 waiter had some customers in his section. Seventeen of them left and the rest of his tables had nine people at each table. He had 5 tables. How many customers did he have initially in his section?<|im_end|>
<|im_start|>assistant
<Z_23><Z_173><Z_104><Z_356><Z_152><Z_163><Z_509><ANSWER>

Question: A waiter had some customers in his section. Seventeen of them left and the rest of his tables had nine people at each table. He had 5 tables. How many customers did he have initially in his section?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Students at Arcadia schools are participating in a coat drive. 9437 coats have been collected so far. Some coats were collected from the high schools, and the rest from the elementary schools which is 2515. How many coats were collected from the high schools?<|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
Students at Arcadia schools are participating in a coat drive. 9437 coats have been collected so far. Some coats were collected from the high schools, and the rest from the elementary schools which is 2515. How many coats were collected from the high schools?<|im_end|>
<|im_start|>assistant
<Z_24><Z_265><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Students at Arcadia schools are participating in a coat drive. 9437 coats have been collected so far. Some coats were collected from the high schools, and the rest from the elementary schools which is 2515. How many coats were collected from the high schools?<|im_end|>
<|im_start|>assistant
<Z_490><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Students at Arcadia schools are participating in a coat drive. 9437 coats have been collected so far. Some coats were collected from the high schools, and the rest from the elementary schools which is 2515. How many coats were collected from the high schools?<|im_end|>
<|im_start|>assistant
<Z_410><Z_428><ANSWER>

Question: Students at Arcadia schools are participating in a coat drive. 9437 coats have been collected so far. Some coats were collected from the high schools, and the rest from the elementary schools which is 2515. How many coats were collected from the high schools?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|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
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|im_end|>
<|im_start|>assistant
<Z_178><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|im_end|>
<|im_start|>assistant
<Z_409><ANSWER>

Question: Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_477><Z_422><Z_264><Z_422><Z_423><Z_233><Z_158><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_283><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_117><Z_268><Z_30><Z_88><Z_153><Z_222><Z_158><ANSWER>

Question: Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_477><Z_400><Z_155><Z_415><Z_211><Z_195><Z_70><Z_358><Z_298><Z_194><Z_219><Z_269><Z_327><Z_79><Z_101><Z_185><Z_393><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_274><Z_40><Z_204><Z_256><Z_152><Z_381><Z_17><Z_126><Z_41><Z_26><Z_477><Z_125><Z_386><Z_299><Z_318><Z_160><Z_225><ANSWER>

Question: A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_375><Z_490><Z_494><Z_439><Z_250><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_361><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_11><Z_258><Z_442><Z_376><Z_219><Z_506><ANSWER>

Question: The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_463><Z_100><Z_138><Z_294><Z_414><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_479><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_142><Z_19><Z_487><Z_477><Z_383><Z_152><Z_204><ANSWER>

Question: What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_24><Z_135><Z_117><Z_220><Z_176><Z_135><Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_349><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_311><Z_483><Z_35><Z_196><Z_316><Z_66><Z_342><ANSWER>

Question: A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|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
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|im_end|>
<|im_start|>assistant
<Z_477><Z_176><Z_366><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|im_end|>
<|im_start|>assistant
<Z_163><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|im_end|>
<|im_start|>assistant
<Z_29><Z_230><Z_354><Z_151><ANSWER>

Question:
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_369><Z_494><Z_440><Z_267><Z_140><Z_53><Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_388><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_96><Z_395><Z_277><Z_95><Z_235><Z_400><Z_204><ANSWER>

Question: A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|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 value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|im_end|>
<|im_start|>assistant
<Z_174><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|im_end|>
<|im_start|>assistant
<Z_283><ANSWER>

Question:
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_24><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_483><Z_75><ANSWER>

Question:
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|im_end|>
<|im_start|>assistant
<Z_369><Z_100><Z_288><Z_267><Z_123><Z_414><Z_159><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|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
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|im_end|>
<|im_start|>assistant
<Z_231><Z_358><Z_401><Z_5><Z_104><Z_130><Z_440><ANSWER>

Question: Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|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
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|im_end|>
<|im_start|>assistant
<Z_477><Z_446><Z_260><Z_437><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|im_end|>
<|im_start|>assistant
<Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|im_end|>
<|im_start|>assistant
<Z_423><Z_174><Z_386><Z_397><Z_326><ANSWER>

Question:
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?


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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_24><Z_135><Z_38><Z_191><Z_200><Z_360><Z_293><Z_240><Z_497><Z_288><Z_305><Z_240><Z_471><Z_456><Z_71><Z_155><ANSWER>

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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_337><ANSWER>

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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_163><Z_103><Z_329><Z_405><Z_308><Z_412><Z_484><Z_204><Z_315><Z_308><Z_125><Z_460><Z_249><Z_347><Z_29><Z_76><ANSWER>

Question:
For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|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
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|im_end|>
<|im_start|>assistant
<Z_477><Z_185><Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|im_end|>
<|im_start|>assistant
<Z_8><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|im_end|>
<|im_start|>assistant
<Z_276><Z_119><Z_67><ANSWER>

Question: Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_215><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_283><ANSWER>

Question: Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|im_end|>
<|im_start|>assistant
<Z_154><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|im_end|>
<|im_start|>assistant
<Z_249><ANSWER>

Question: A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventeen is the seventh prime number. What is the twentieth prime number?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventeen is the seventh prime number. What is the twentieth prime number?<|im_end|>
<|im_start|>assistant
<Z_511><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventeen is the seventh prime number. What is the twentieth prime number?<|im_end|>
<|im_start|>assistant
<Z_311><ANSWER>

Question: Seventeen is the seventh prime number. What is the twentieth prime number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_300><Z_377><Z_465><Z_255><Z_192><Z_77><Z_460><Z_238><Z_348><Z_378><Z_443><Z_488><Z_490><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_270><Z_137><Z_35><Z_50><Z_57><Z_78><Z_91><Z_194><Z_451><Z_406><Z_382><Z_276><Z_438><Z_476><ANSWER>

Question: Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_369><Z_84><Z_214><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_239><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_266><Z_177><Z_398><ANSWER>

Question: In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_477><Z_294><Z_54><Z_346><Z_478><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_450><Z_108><Z_403><Z_426><Z_145><Z_204><ANSWER>

Question: There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|im_end|>
<|im_start|>assistant
<Z_477><Z_446><Z_294><Z_113><Z_391><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|im_end|>
<|im_start|>assistant
<Z_419><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|im_end|>
<|im_start|>assistant
<Z_117><Z_19><Z_201><Z_302><Z_39><ANSWER>

Question: Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_369><Z_467><Z_119><Z_328><Z_214><Z_418><Z_374><Z_456><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|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
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_217><Z_472><Z_57><Z_36><Z_443><Z_154><Z_362><Z_213><ANSWER>

Question: Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_322><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_189><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_16><Z_39><Z_175><ANSWER>

Question: How many even divisors of $8!$ are also divisible by $3$?