<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequences ${a_{n}}$ and ${b_{n}}$, $a_{n+1}=a_{n}+b_{n}+ \sqrt {a_{n}^{2}+b_{n}^{2}}$, $b_{n+1}=a_{n}+b_{n}- \sqrt {a_{n}^{2}+b_{n}^{2}}$, $a_{1}=1$, $b_{1}=1$. Let $c_{n}= \frac {1}{a_{n}}+ \frac {1}{b_{n}}$, then the sum of the first $2017$ terms of the sequence ${c_{n}}$ is ___.<|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 sequences ${a_{n}}$ and ${b_{n}}$, $a_{n+1}=a_{n}+b_{n}+ \sqrt {a_{n}^{2}+b_{n}^{2}}$, $b_{n+1}=a_{n}+b_{n}- \sqrt {a_{n}^{2}+b_{n}^{2}}$, $a_{1}=1$, $b_{1}=1$. Let $c_{n}= \frac {1}{a_{n}}+ \frac {1}{b_{n}}$, then the sum of the first $2017$ terms of the sequence ${c_{n}}$ is ___.<|im_end|>
<|im_start|>assistant
<Z_24><Z_245><Z_391><Z_154><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequences ${a_{n}}$ and ${b_{n}}$, $a_{n+1}=a_{n}+b_{n}+ \sqrt {a_{n}^{2}+b_{n}^{2}}$, $b_{n+1}=a_{n}+b_{n}- \sqrt {a_{n}^{2}+b_{n}^{2}}$, $a_{1}=1$, $b_{1}=1$. Let $c_{n}= \frac {1}{a_{n}}+ \frac {1}{b_{n}}$, then the sum of the first $2017$ terms of the sequence ${c_{n}}$ is ___.<|im_end|>
<|im_start|>assistant
<Z_378><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequences ${a_{n}}$ and ${b_{n}}$, $a_{n+1}=a_{n}+b_{n}+ \sqrt {a_{n}^{2}+b_{n}^{2}}$, $b_{n+1}=a_{n}+b_{n}- \sqrt {a_{n}^{2}+b_{n}^{2}}$, $a_{1}=1$, $b_{1}=1$. Let $c_{n}= \frac {1}{a_{n}}+ \frac {1}{b_{n}}$, then the sum of the first $2017$ terms of the sequence ${c_{n}}$ is ___.<|im_end|>
<|im_start|>assistant
<Z_177><Z_256><Z_182><Z_489><ANSWER>

Question: In the sequences ${a_{n}}$ and ${b_{n}}$, $a_{n+1}=a_{n}+b_{n}+ \sqrt {a_{n}^{2}+b_{n}^{2}}$, $b_{n+1}=a_{n}+b_{n}- \sqrt {a_{n}^{2}+b_{n}^{2}}$, $a_{1}=1$, $b_{1}=1$. Let $c_{n}= \frac {1}{a_{n}}+ \frac {1}{b_{n}}$, then the sum of the first $2017$ terms of the sequence ${c_{n}}$ is ___.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last year, 90171 people were born in a country, and 16320 people immigrated to the country. However, 8212 people emigrated from the country and the death rate was 0.08% of the total population which stood at 2876543 at the beginning of the year. Calculate the net increase in population for that year.<|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
Last year, 90171 people were born in a country, and 16320 people immigrated to the country. However, 8212 people emigrated from the country and the death rate was 0.08% of the total population which stood at 2876543 at the beginning of the year. Calculate the net increase in population for that year.<|im_end|>
<|im_start|>assistant
<Z_477><Z_494><Z_509><Z_172><Z_272><Z_172><Z_67><Z_432><Z_236><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last year, 90171 people were born in a country, and 16320 people immigrated to the country. However, 8212 people emigrated from the country and the death rate was 0.08% of the total population which stood at 2876543 at the beginning of the year. Calculate the net increase in population for that year.<|im_end|>
<|im_start|>assistant
<Z_417><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last year, 90171 people were born in a country, and 16320 people immigrated to the country. However, 8212 people emigrated from the country and the death rate was 0.08% of the total population which stood at 2876543 at the beginning of the year. Calculate the net increase in population for that year.<|im_end|>
<|im_start|>assistant
<Z_220><Z_508><Z_369><Z_498><Z_8><Z_383><Z_481><Z_142><Z_118><ANSWER>

Question: Last year, 90171 people were born in a country, and 16320 people immigrated to the country. However, 8212 people emigrated from the country and the death rate was 0.08% of the total population which stood at 2876543 at the beginning of the year. Calculate the net increase in population for that year.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive whole numbers have cube roots that are less than $15$? For example, $3000$ would count since $\sqrt[3]{3000}<15.$<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive whole numbers have cube roots that are less than $15$? For example, $3000$ would count since $\sqrt[3]{3000}<15.$<|im_end|>
<|im_start|>assistant
<Z_477><Z_210><Z_278><Z_238><Z_464><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive whole numbers have cube roots that are less than $15$? For example, $3000$ would count since $\sqrt[3]{3000}<15.$<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive whole numbers have cube roots that are less than $15$? For example, $3000$ would count since $\sqrt[3]{3000}<15.$<|im_end|>
<|im_start|>assistant
<Z_308><Z_58><Z_486><Z_109><Z_500><Z_201><ANSWER>

Question: How many positive whole numbers have cube roots that are less than $15$? For example, $3000$ would count since $\sqrt[3]{3000}<15.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?<|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
Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?<|im_end|>
<|im_start|>assistant
<Z_369><Z_392><Z_144><Z_140><Z_124><Z_389><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?<|im_end|>
<|im_start|>assistant
<Z_133><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?<|im_end|>
<|im_start|>assistant
<Z_436><Z_393><Z_431><Z_72><Z_168><Z_74><ANSWER>

Question: Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some men take 21 days of 8 hours each to do a piece of work. It takes 60 days of 3 hours each for 21 women to do the same work, given that 3 women do as much work as 2 men. How many men were working on the piece of work?<|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
Some men take 21 days of 8 hours each to do a piece of work. It takes 60 days of 3 hours each for 21 women to do the same work, given that 3 women do as much work as 2 men. How many men were working on the piece of work?<|im_end|>
<|im_start|>assistant
<Z_24><Z_244><Z_362><Z_223><Z_308><Z_59><Z_185><Z_187><Z_173><Z_372><Z_393><Z_391><Z_313><Z_456><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some men take 21 days of 8 hours each to do a piece of work. It takes 60 days of 3 hours each for 21 women to do the same work, given that 3 women do as much work as 2 men. How many men were working on the piece of work?<|im_end|>
<|im_start|>assistant
<Z_204><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some men take 21 days of 8 hours each to do a piece of work. It takes 60 days of 3 hours each for 21 women to do the same work, given that 3 women do as much work as 2 men. How many men were working on the piece of work?<|im_end|>
<|im_start|>assistant
<Z_372><Z_348><Z_465><Z_43><Z_438><Z_100><Z_222><Z_369><Z_229><Z_6><Z_266><Z_201><Z_145><Z_489><Z_430><ANSWER>

Question: Some men take 21 days of 8 hours each to do a piece of work. It takes 60 days of 3 hours each for 21 women to do the same work, given that 3 women do as much work as 2 men. How many men were working on the piece of work?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?<|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
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?<|im_end|>
<|im_start|>assistant
<Z_369><Z_73><Z_87><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?<|im_end|>
<|im_start|>assistant
<Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?<|im_end|>
<|im_start|>assistant
<Z_431><Z_90><Z_99><Z_356><ANSWER>

Question: 0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle, let the  $A$ -altitude meet  $BC$  at  $D$ , let the  $B$ -altitude meet  $AC$  at  $E$ , and let  $T\neq A$  be the point on the circumcircle of  $ABC$  such that  $AT || BC$ . Given that  $D,E,T$  are collinear, if  $BD=3$  and  $AD=4$ , then the area of  $ABC$  can be written as  $a+\sqrt{b}$ , where  $a$  and  $b$  are positive integers. What is  $a+b$ ?

*2021 CCA Math Bonanza Individual Round #12*<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle, let the  $A$ -altitude meet  $BC$  at  $D$ , let the  $B$ -altitude meet  $AC$  at  $E$ , and let  $T\neq A$  be the point on the circumcircle of  $ABC$  such that  $AT || BC$ . Given that  $D,E,T$  are collinear, if  $BD=3$  and  $AD=4$ , then the area of  $ABC$  can be written as  $a+\sqrt{b}$ , where  $a$  and  $b$  are positive integers. What is  $a+b$ ?

*2021 CCA Math Bonanza Individual Round #12*<|im_end|>
<|im_start|>assistant
<Z_24><Z_335><Z_395><Z_83><Z_472><Z_229><Z_250><Z_67><Z_135><Z_102><Z_460><Z_53><Z_59><Z_366><Z_236><Z_121><Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle, let the  $A$ -altitude meet  $BC$  at  $D$ , let the  $B$ -altitude meet  $AC$  at  $E$ , and let  $T\neq A$  be the point on the circumcircle of  $ABC$  such that  $AT || BC$ . Given that  $D,E,T$  are collinear, if  $BD=3$  and  $AD=4$ , then the area of  $ABC$  can be written as  $a+\sqrt{b}$ , where  $a$  and  $b$  are positive integers. What is  $a+b$ ?

*2021 CCA Math Bonanza Individual Round #12*<|im_end|>
<|im_start|>assistant
<Z_186><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle, let the  $A$ -altitude meet  $BC$  at  $D$ , let the  $B$ -altitude meet  $AC$  at  $E$ , and let  $T\neq A$  be the point on the circumcircle of  $ABC$  such that  $AT || BC$ . Given that  $D,E,T$  are collinear, if  $BD=3$  and  $AD=4$ , then the area of  $ABC$  can be written as  $a+\sqrt{b}$ , where  $a$  and  $b$  are positive integers. What is  $a+b$ ?

*2021 CCA Math Bonanza Individual Round #12*<|im_end|>
<|im_start|>assistant
<Z_195><Z_326><Z_227><Z_312><Z_227><Z_156><Z_75><Z_21><Z_272><Z_321><Z_153><Z_466><Z_45><Z_394><Z_12><Z_131><Z_68><ANSWER>

Question:
Let  $ABC$  be a triangle, let the  $A$ -altitude meet  $BC$  at  $D$ , let the  $B$ -altitude meet  $AC$  at  $E$ , and let  $T\neq A$  be the point on the circumcircle of  $ABC$  such that  $AT || BC$ . Given that  $D,E,T$  are collinear, if  $BD=3$  and  $AD=4$ , then the area of  $ABC$  can be written as  $a+\sqrt{b}$ , where  $a$  and  $b$  are positive integers. What is  $a+b$ ?

*2021 CCA Math Bonanza Individual Round #12*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The point P is equidistant from point A (1,0) and the line $x=-1$, and the distance from point P to the line $y=x$ is $\frac{\sqrt{2}}{2}$. Determine the number of such points P.<|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 point P is equidistant from point A (1,0) and the line $x=-1$, and the distance from point P to the line $y=x$ is $\frac{\sqrt{2}}{2}$. Determine the number of such points P.<|im_end|>
<|im_start|>assistant
<Z_369><Z_106><Z_420><Z_437><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The point P is equidistant from point A (1,0) and the line $x=-1$, and the distance from point P to the line $y=x$ is $\frac{\sqrt{2}}{2}$. Determine the number of such points P.<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The point P is equidistant from point A (1,0) and the line $x=-1$, and the distance from point P to the line $y=x$ is $\frac{\sqrt{2}}{2}$. Determine the number of such points P.<|im_end|>
<|im_start|>assistant
<Z_163><Z_190><Z_394><Z_164><ANSWER>

Question: The point P is equidistant from point A (1,0) and the line $x=-1$, and the distance from point P to the line $y=x$ is $\frac{\sqrt{2}}{2}$. Determine the number of such points P.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 3 different math books, 4 different physics books, and 2 different chemistry books.
(Ⅰ) If all books are to be arranged on the same shelf without separating books of the same subject, how many different arrangements are there?
(Ⅱ) If selecting any 5 books from them, calculate:
① exactly 2 math books, 2 physics books, and 1 chemistry book, how many different ways of selection?
② at least 1 math book, how many different ways of selection?<|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 are 3 different math books, 4 different physics books, and 2 different chemistry books.
(Ⅰ) If all books are to be arranged on the same shelf without separating books of the same subject, how many different arrangements are there?
(Ⅱ) If selecting any 5 books from them, calculate:
① exactly 2 math books, 2 physics books, and 1 chemistry book, how many different ways of selection?
② at least 1 math book, how many different ways of selection?<|im_end|>
<|im_start|>assistant
<Z_477><Z_314><Z_459><Z_205><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 3 different math books, 4 different physics books, and 2 different chemistry books.
(Ⅰ) If all books are to be arranged on the same shelf without separating books of the same subject, how many different arrangements are there?
(Ⅱ) If selecting any 5 books from them, calculate:
① exactly 2 math books, 2 physics books, and 1 chemistry book, how many different ways of selection?
② at least 1 math book, how many different ways of selection?<|im_end|>
<|im_start|>assistant
<Z_46><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 3 different math books, 4 different physics books, and 2 different chemistry books.
(Ⅰ) If all books are to be arranged on the same shelf without separating books of the same subject, how many different arrangements are there?
(Ⅱ) If selecting any 5 books from them, calculate:
① exactly 2 math books, 2 physics books, and 1 chemistry book, how many different ways of selection?
② at least 1 math book, how many different ways of selection?<|im_end|>
<|im_start|>assistant
<Z_488><Z_360><Z_30><Z_354><Z_317><ANSWER>

Question:
There are 3 different math books, 4 different physics books, and 2 different chemistry books.
(Ⅰ) If all books are to be arranged on the same shelf without separating books of the same subject, how many different arrangements are there?
(Ⅱ) If selecting any 5 books from them, calculate:
① exactly 2 math books, 2 physics books, and 1 chemistry book, how many different ways of selection?
② at least 1 math book, how many different ways of selection?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the equilateral triangle $\triangle ABC$ with side length $4$, $D$ is the midpoint of $BC$. Calculate the dot product $\overrightarrow{DA} \cdot \overrightarrow{AB} =$ _____.<|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 the equilateral triangle $\triangle ABC$ with side length $4$, $D$ is the midpoint of $BC$. Calculate the dot product $\overrightarrow{DA} \cdot \overrightarrow{AB} =$ _____.<|im_end|>
<|im_start|>assistant
<Z_477><Z_157><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the equilateral triangle $\triangle ABC$ with side length $4$, $D$ is the midpoint of $BC$. Calculate the dot product $\overrightarrow{DA} \cdot \overrightarrow{AB} =$ _____.<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the equilateral triangle $\triangle ABC$ with side length $4$, $D$ is the midpoint of $BC$. Calculate the dot product $\overrightarrow{DA} \cdot \overrightarrow{AB} =$ _____.<|im_end|>
<|im_start|>assistant
<Z_346><Z_194><Z_119><ANSWER>

Question: In the equilateral triangle $\triangle ABC$ with side length $4$, $D$ is the midpoint of $BC$. Calculate the dot product $\overrightarrow{DA} \cdot \overrightarrow{AB} =$ _____.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$?<|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 equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_403><Z_227><Z_143><Z_434><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$?<|im_end|>
<|im_start|>assistant
<Z_190><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$?<|im_end|>
<|im_start|>assistant
<Z_322><Z_418><Z_382><Z_286><Z_17><ANSWER>

Question: The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julian is writing a comic book. His story has 143 frames in all. If he wants to put a certain number of frames on each page, he would have 13 pages. How many frames does he want to put on each page?<|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
Julian is writing a comic book. His story has 143 frames in all. If he wants to put a certain number of frames on each page, he would have 13 pages. How many frames does he want to put on each page?<|im_end|>
<|im_start|>assistant
<Z_477><Z_61><Z_43><Z_497><Z_178><Z_441><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julian is writing a comic book. His story has 143 frames in all. If he wants to put a certain number of frames on each page, he would have 13 pages. How many frames does he want to put on each page?<|im_end|>
<|im_start|>assistant
<Z_200><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julian is writing a comic book. His story has 143 frames in all. If he wants to put a certain number of frames on each page, he would have 13 pages. How many frames does he want to put on each page?<|im_end|>
<|im_start|>assistant
<Z_430><Z_101><Z_423><Z_259><Z_379><Z_482><Z_373><ANSWER>

Question: Julian is writing a comic book. His story has 143 frames in all. If he wants to put a certain number of frames on each page, he would have 13 pages. How many frames does he want to put on each page?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of $a$ such that $ax^2 + 8x + 16$ is the square of a binomial.<|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 $a$ such that $ax^2 + 8x + 16$ is the square of a binomial.<|im_end|>
<|im_start|>assistant
<Z_369><Z_323><Z_394><Z_299><Z_494><Z_471><Z_413><Z_123><Z_312><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 $a$ such that $ax^2 + 8x + 16$ is the square of a binomial.<|im_end|>
<|im_start|>assistant
<Z_482><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of $a$ such that $ax^2 + 8x + 16$ is the square of a binomial.<|im_end|>
<|im_start|>assistant
<Z_13><Z_290><Z_385><Z_102><Z_431><Z_482><Z_484><Z_213><Z_16><ANSWER>

Question: Find the value of $a$ such that $ax^2 + 8x + 16$ is the square of a binomial.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many lines can be drawn through a point given inside a given triangle that will intersect similar triangles from the given triangle?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many lines can be drawn through a point given inside a given triangle that will intersect similar triangles from the given triangle?<|im_end|>
<|im_start|>assistant
<Z_369><Z_414><Z_343><Z_176><Z_357><Z_197><Z_322><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many lines can be drawn through a point given inside a given triangle that will intersect similar triangles from the given triangle?<|im_end|>
<|im_start|>assistant
<Z_30><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many lines can be drawn through a point given inside a given triangle that will intersect similar triangles from the given triangle?<|im_end|>
<|im_start|>assistant
<Z_477><Z_344><Z_89><Z_221><Z_308><Z_313><Z_398><Z_346><ANSWER>

Question: How many lines can be drawn through a point given inside a given triangle that will intersect similar triangles from the given triangle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit positive integers are congruent to 2 (mod 4)?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit positive integers are congruent to 2 (mod 4)?<|im_end|>
<|im_start|>assistant
<Z_477><Z_39><Z_134><Z_77><Z_134><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit positive integers are congruent to 2 (mod 4)?<|im_end|>
<|im_start|>assistant
<Z_271><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit positive integers are congruent to 2 (mod 4)?<|im_end|>
<|im_start|>assistant
<Z_234><Z_29><Z_139><Z_155><Z_157><ANSWER>

Question: How many three-digit positive integers are congruent to 2 (mod 4)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class, 6 students can speak Gujarati, some students can speak Hindi, and 6 can speak Marathi. Two students can speak two languages, and one student can speak all three languages. There are 22 students in the class. How many students can speak Hindi?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class, 6 students can speak Gujarati, some students can speak Hindi, and 6 can speak Marathi. Two students can speak two languages, and one student can speak all three languages. There are 22 students in the class. How many students can speak Hindi?<|im_end|>
<|im_start|>assistant
<Z_24><Z_255><Z_48><Z_171><Z_139><Z_432><Z_185><Z_510><Z_414><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class, 6 students can speak Gujarati, some students can speak Hindi, and 6 can speak Marathi. Two students can speak two languages, and one student can speak all three languages. There are 22 students in the class. How many students can speak Hindi?<|im_end|>
<|im_start|>assistant
<Z_464><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class, 6 students can speak Gujarati, some students can speak Hindi, and 6 can speak Marathi. Two students can speak two languages, and one student can speak all three languages. There are 22 students in the class. How many students can speak Hindi?<|im_end|>
<|im_start|>assistant
<Z_491><Z_407><Z_335><Z_57><Z_44><Z_237><Z_155><Z_79><Z_284><ANSWER>

Question: In a class, 6 students can speak Gujarati, some students can speak Hindi, and 6 can speak Marathi. Two students can speak two languages, and one student can speak all three languages. There are 22 students in the class. How many students can speak Hindi?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Can you arrange 9 digits in the form of a square in such a way that in any row, any column, and on each of the main diagonals, the difference between the sum of two digits and the third digit is the same across all these lines? How many such solutions exist?

Note: In the provided example square, all rows and columns meet the required condition — the difference is 3 (e.g., \(4 + 2 - 3\), \(1 + 9 - 7\), \(6 + 5 - 8\), etc.). However, the diagonals do not satisfy the condition because the differences \(8 - (4 + 1)\) and \(6 - (1 + 2)\) are calculated incorrectly. The correct approach should involve subtracting one digit from the sum of the other two.<|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
Can you arrange 9 digits in the form of a square in such a way that in any row, any column, and on each of the main diagonals, the difference between the sum of two digits and the third digit is the same across all these lines? How many such solutions exist?

Note: In the provided example square, all rows and columns meet the required condition — the difference is 3 (e.g., \(4 + 2 - 3\), \(1 + 9 - 7\), \(6 + 5 - 8\), etc.). However, the diagonals do not satisfy the condition because the differences \(8 - (4 + 1)\) and \(6 - (1 + 2)\) are calculated incorrectly. The correct approach should involve subtracting one digit from the sum of the other two.<|im_end|>
<|im_start|>assistant
<Z_24><Z_483><Z_343><Z_292><Z_321><Z_188><Z_94><Z_362><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Can you arrange 9 digits in the form of a square in such a way that in any row, any column, and on each of the main diagonals, the difference between the sum of two digits and the third digit is the same across all these lines? How many such solutions exist?

Note: In the provided example square, all rows and columns meet the required condition — the difference is 3 (e.g., \(4 + 2 - 3\), \(1 + 9 - 7\), \(6 + 5 - 8\), etc.). However, the diagonals do not satisfy the condition because the differences \(8 - (4 + 1)\) and \(6 - (1 + 2)\) are calculated incorrectly. The correct approach should involve subtracting one digit from the sum of the other two.<|im_end|>
<|im_start|>assistant
<Z_397><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Can you arrange 9 digits in the form of a square in such a way that in any row, any column, and on each of the main diagonals, the difference between the sum of two digits and the third digit is the same across all these lines? How many such solutions exist?

Note: In the provided example square, all rows and columns meet the required condition — the difference is 3 (e.g., \(4 + 2 - 3\), \(1 + 9 - 7\), \(6 + 5 - 8\), etc.). However, the diagonals do not satisfy the condition because the differences \(8 - (4 + 1)\) and \(6 - (1 + 2)\) are calculated incorrectly. The correct approach should involve subtracting one digit from the sum of the other two.<|im_end|>
<|im_start|>assistant
<Z_95><Z_318><Z_128><Z_30><Z_84><Z_414><Z_134><Z_411><ANSWER>

Question:
Can you arrange 9 digits in the form of a square in such a way that in any row, any column, and on each of the main diagonals, the difference between the sum of two digits and the third digit is the same across all these lines? How many such solutions exist?

Note: In the provided example square, all rows and columns meet the required condition — the difference is 3 (e.g., \(4 + 2 - 3\), \(1 + 9 - 7\), \(6 + 5 - 8\), etc.). However, the diagonals do not satisfy the condition because the differences \(8 - (4 + 1)\) and \(6 - (1 + 2)\) are calculated incorrectly. The correct approach should involve subtracting one digit from the sum of the other two.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.<|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 the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_156><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.<|im_end|>
<|im_start|>assistant
<Z_443><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.<|im_end|>
<|im_start|>assistant
<Z_433><Z_362><Z_168><ANSWER>

Question: Given the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $P_{1}: y=x^{2}+\frac{203}{200}$ and $P_{2}: x=y^{2}+\frac{27}{2}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $P_{1}: y=x^{2}+\frac{203}{200}$ and $P_{2}: x=y^{2}+\frac{27}{2}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_200><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $P_{1}: y=x^{2}+\frac{203}{200}$ and $P_{2}: x=y^{2}+\frac{27}{2}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_237><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $P_{1}: y=x^{2}+\frac{203}{200}$ and $P_{2}: x=y^{2}+\frac{27}{2}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_499><Z_369><Z_26><ANSWER>

Question: Let $P_{1}: y=x^{2}+\frac{203}{200}$ and $P_{2}: x=y^{2}+\frac{27}{2}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ten gangsters are standing in a field. The distance between each pair of gangsters is different. When the clock strikes, each gangster shoots the nearest gangster dead. What is the largest number of gangsters that can survive?<|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
Ten gangsters are standing in a field. The distance between each pair of gangsters is different. When the clock strikes, each gangster shoots the nearest gangster dead. What is the largest number of gangsters that can survive?<|im_end|>
<|im_start|>assistant
<Z_24><Z_160><Z_250><Z_381><Z_346><Z_335><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ten gangsters are standing in a field. The distance between each pair of gangsters is different. When the clock strikes, each gangster shoots the nearest gangster dead. What is the largest number of gangsters that can survive?<|im_end|>
<|im_start|>assistant
<Z_168><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ten gangsters are standing in a field. The distance between each pair of gangsters is different. When the clock strikes, each gangster shoots the nearest gangster dead. What is the largest number of gangsters that can survive?<|im_end|>
<|im_start|>assistant
<Z_65><Z_501><Z_449><Z_255><Z_457><Z_317><ANSWER>

Question: Ten gangsters are standing in a field. The distance between each pair of gangsters is different. When the clock strikes, each gangster shoots the nearest gangster dead. What is the largest number of gangsters that can survive?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Every bag of Sweetsies (a type of candy) contains the same number of candies.

When the candies from one bag are shared equally among $8$ children, $5$ candies remain.

If the candies from three bags are then shared equally among these $8$ children, what is the smallest number of candies that could possibly be left over?<|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
Every bag of Sweetsies (a type of candy) contains the same number of candies.

When the candies from one bag are shared equally among $8$ children, $5$ candies remain.

If the candies from three bags are then shared equally among these $8$ children, what is the smallest number of candies that could possibly be left over?<|im_end|>
<|im_start|>assistant
<Z_24><Z_392><Z_406><Z_245><Z_50><Z_84><Z_197><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Every bag of Sweetsies (a type of candy) contains the same number of candies.

When the candies from one bag are shared equally among $8$ children, $5$ candies remain.

If the candies from three bags are then shared equally among these $8$ children, what is the smallest number of candies that could possibly be left over?<|im_end|>
<|im_start|>assistant
<Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Every bag of Sweetsies (a type of candy) contains the same number of candies.

When the candies from one bag are shared equally among $8$ children, $5$ candies remain.

If the candies from three bags are then shared equally among these $8$ children, what is the smallest number of candies that could possibly be left over?<|im_end|>
<|im_start|>assistant
<Z_202><Z_177><Z_160><Z_15><Z_278><Z_238><Z_3><Z_259><ANSWER>

Question:
Every bag of Sweetsies (a type of candy) contains the same number of candies.

When the candies from one bag are shared equally among $8$ children, $5$ candies remain.

If the candies from three bags are then shared equally among these $8$ children, what is the smallest number of candies that could possibly be left over?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An investment of $\$$8,000 is made in a government bond that will pay 5$\%$ interest compounded monthly. At the end of 3 years, what is the total amount of money in this investment?<|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
An investment of $\$$8,000 is made in a government bond that will pay 5$\%$ interest compounded monthly. At the end of 3 years, what is the total amount of money in this investment?<|im_end|>
<|im_start|>assistant
<Z_369><Z_307><Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An investment of $\$$8,000 is made in a government bond that will pay 5$\%$ interest compounded monthly. At the end of 3 years, what is the total amount of money in this investment?<|im_end|>
<|im_start|>assistant
<Z_229><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An investment of $\$$8,000 is made in a government bond that will pay 5$\%$ interest compounded monthly. At the end of 3 years, what is the total amount of money in this investment?<|im_end|>
<|im_start|>assistant
<Z_217><Z_208><Z_22><ANSWER>

Question: An investment of $\$$8,000 is made in a government bond that will pay 5$\%$ interest compounded monthly. At the end of 3 years, what is the total amount of money in this investment?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a piece of work in 10 days and b in 20 days . they began the work together but 5 days before the completion of the work , a leaves . the work was completed in ?<|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 can do a piece of work in 10 days and b in 20 days . they began the work together but 5 days before the completion of the work , a leaves . the work was completed in ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_146><Z_42><Z_377><Z_502><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a piece of work in 10 days and b in 20 days . they began the work together but 5 days before the completion of the work , a leaves . the work was completed in ?<|im_end|>
<|im_start|>assistant
<Z_387><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a piece of work in 10 days and b in 20 days . they began the work together but 5 days before the completion of the work , a leaves . the work was completed in ?<|im_end|>
<|im_start|>assistant
<Z_107><Z_324><Z_269><Z_177><Z_23><Z_337><ANSWER>

Question: a can do a piece of work in 10 days and b in 20 days . they began the work together but 5 days before the completion of the work , a leaves . the work was completed in ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one-eighth of $2^{40}$ is equal to $2^x$, what is $x$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one-eighth of $2^{40}$ is equal to $2^x$, what is $x$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_209><Z_378><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one-eighth of $2^{40}$ is equal to $2^x$, what is $x$?<|im_end|>
<|im_start|>assistant
<Z_157><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one-eighth of $2^{40}$ is equal to $2^x$, what is $x$?<|im_end|>
<|im_start|>assistant
<Z_218><Z_368><Z_39><ANSWER>

Question: If one-eighth of $2^{40}$ is equal to $2^x$, what is $x$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain solution of alcohol and water is 5 percent alcohol. If 4.5 liters of alcohol and 5.5 liters of water are added to this solution, the solution produced is 13 percent alcohol. What is the initial volume of the solution?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain solution of alcohol and water is 5 percent alcohol. If 4.5 liters of alcohol and 5.5 liters of water are added to this solution, the solution produced is 13 percent alcohol. What is the initial volume of the solution?<|im_end|>
<|im_start|>assistant
<Z_369><Z_228><Z_161><Z_146><Z_422><Z_313><Z_219><Z_200><Z_217><Z_94><Z_294><Z_171><Z_509><Z_37><Z_470><Z_144><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain solution of alcohol and water is 5 percent alcohol. If 4.5 liters of alcohol and 5.5 liters of water are added to this solution, the solution produced is 13 percent alcohol. What is the initial volume of the solution?<|im_end|>
<|im_start|>assistant
<Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain solution of alcohol and water is 5 percent alcohol. If 4.5 liters of alcohol and 5.5 liters of water are added to this solution, the solution produced is 13 percent alcohol. What is the initial volume of the solution?<|im_end|>
<|im_start|>assistant
<Z_242><Z_463><Z_306><Z_352><Z_3><Z_134><Z_68><Z_244><Z_382><Z_369><Z_369><Z_140><Z_480><Z_120><Z_249><Z_38><ANSWER>

Question: A certain solution of alcohol and water is 5 percent alcohol. If 4.5 liters of alcohol and 5.5 liters of water are added to this solution, the solution produced is 13 percent alcohol. What is the initial volume of the solution?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle DEF$, with vertices $D$ at the origin, $E$ at $(8,0)$, and $F$ at $(2,4)$, determine the sum of the distances from the point $Q = (3,1)$ to each vertex of the triangle. Express the sum in the form $a\sqrt{p} + b\sqrt{q}$, where $a$, $b$, $p$, $q$ are integers, and find the value of $a + b$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle DEF$, with vertices $D$ at the origin, $E$ at $(8,0)$, and $F$ at $(2,4)$, determine the sum of the distances from the point $Q = (3,1)$ to each vertex of the triangle. Express the sum in the form $a\sqrt{p} + b\sqrt{q}$, where $a$, $b$, $p$, $q$ are integers, and find the value of $a + b$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle DEF$, with vertices $D$ at the origin, $E$ at $(8,0)$, and $F$ at $(2,4)$, determine the sum of the distances from the point $Q = (3,1)$ to each vertex of the triangle. Express the sum in the form $a\sqrt{p} + b\sqrt{q}$, where $a$, $b$, $p$, $q$ are integers, and find the value of $a + b$.<|im_end|>
<|im_start|>assistant
<Z_200><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle DEF$, with vertices $D$ at the origin, $E$ at $(8,0)$, and $F$ at $(2,4)$, determine the sum of the distances from the point $Q = (3,1)$ to each vertex of the triangle. Express the sum in the form $a\sqrt{p} + b\sqrt{q}$, where $a$, $b$, $p$, $q$ are integers, and find the value of $a + b$.<|im_end|>
<|im_start|>assistant
<Z_135><Z_315><ANSWER>

Question: In triangle $\triangle DEF$, with vertices $D$ at the origin, $E$ at $(8,0)$, and $F$ at $(2,4)$, determine the sum of the distances from the point $Q = (3,1)$ to each vertex of the triangle. Express the sum in the form $a\sqrt{p} + b\sqrt{q}$, where $a$, $b$, $p$, $q$ are integers, and find the value of $a + b$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At a gathering, there are 25 gremlins and 20 imps. Owing to some conflicts, only 10 imps are willing to shake hands with the gremlins, and each of these imps shakes hands with just 15 of the gremlins. All gremlins are in good terms with each other and shake hands amongst themselves as well. Calculate the total number of handshakes at this assembly.<|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
At a gathering, there are 25 gremlins and 20 imps. Owing to some conflicts, only 10 imps are willing to shake hands with the gremlins, and each of these imps shakes hands with just 15 of the gremlins. All gremlins are in good terms with each other and shake hands amongst themselves as well. Calculate the total number of handshakes at this assembly.<|im_end|>
<|im_start|>assistant
<Z_477><Z_443><Z_210><Z_318><Z_374><Z_87><Z_361><Z_288><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At a gathering, there are 25 gremlins and 20 imps. Owing to some conflicts, only 10 imps are willing to shake hands with the gremlins, and each of these imps shakes hands with just 15 of the gremlins. All gremlins are in good terms with each other and shake hands amongst themselves as well. Calculate the total number of handshakes at this assembly.<|im_end|>
<|im_start|>assistant
<Z_105><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At a gathering, there are 25 gremlins and 20 imps. Owing to some conflicts, only 10 imps are willing to shake hands with the gremlins, and each of these imps shakes hands with just 15 of the gremlins. All gremlins are in good terms with each other and shake hands amongst themselves as well. Calculate the total number of handshakes at this assembly.<|im_end|>
<|im_start|>assistant
<Z_71><Z_2><Z_408><Z_252><Z_237><Z_69><Z_422><Z_207><Z_2><ANSWER>

Question: At a gathering, there are 25 gremlins and 20 imps. Owing to some conflicts, only 10 imps are willing to shake hands with the gremlins, and each of these imps shakes hands with just 15 of the gremlins. All gremlins are in good terms with each other and shake hands amongst themselves as well. Calculate the total number of handshakes at this assembly.