<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?<|im_end|>
<|im_start|>assistant
<Z_104><Z_120><Z_47><Z_125><Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?<|im_end|>
<|im_start|>assistant
<Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?<|im_end|>
<|im_start|>assistant
<Z_8><Z_113><Z_79><Z_42><Z_10><ANSWER>

Question: If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One hundred bricks, each measuring $3''\times11''\times18''$, are to be stacked one on top of another in a tower. Each brick can be oriented to contribute $3''$, $11''$, or $18''$ to the tower height. Determine how many different tower heights can be achieved using all one hundred bricks.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One hundred bricks, each measuring $3''\times11''\times18''$, are to be stacked one on top of another in a tower. Each brick can be oriented to contribute $3''$, $11''$, or $18''$ to the tower height. Determine how many different tower heights can be achieved using all one hundred bricks.<|im_end|>
<|im_start|>assistant
<Z_27><Z_105><Z_83><Z_88><Z_75><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One hundred bricks, each measuring $3''\times11''\times18''$, are to be stacked one on top of another in a tower. Each brick can be oriented to contribute $3''$, $11''$, or $18''$ to the tower height. Determine how many different tower heights can be achieved using all one hundred bricks.<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One hundred bricks, each measuring $3''\times11''\times18''$, are to be stacked one on top of another in a tower. Each brick can be oriented to contribute $3''$, $11''$, or $18''$ to the tower height. Determine how many different tower heights can be achieved using all one hundred bricks.<|im_end|>
<|im_start|>assistant
<Z_36><Z_117><Z_43><Z_85><Z_14><Z_78><ANSWER>

Question: One hundred bricks, each measuring $3''\times11''\times18''$, are to be stacked one on top of another in a tower. Each brick can be oriented to contribute $3''$, $11''$, or $18''$ to the tower height. Determine how many different tower heights can be achieved using all one hundred bricks.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_103><Z_43><Z_125><Z_115><Z_56><Z_22><Z_39><Z_70><Z_19><Z_58><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.<|im_end|>
<|im_start|>assistant
<Z_72><Z_22><Z_127><Z_77><Z_57><Z_64><Z_29><Z_50><Z_60><Z_90><Z_123><Z_33><ANSWER>

Question: Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given: $x : y : z = 1 : 2 : 7$ and $2x - y + 3z = 105$, find the value of $xyz$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given: $x : y : z = 1 : 2 : 7$ and $2x - y + 3z = 105$, find the value of $xyz$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_77><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given: $x : y : z = 1 : 2 : 7$ and $2x - y + 3z = 105$, find the value of $xyz$.<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given: $x : y : z = 1 : 2 : 7$ and $2x - y + 3z = 105$, find the value of $xyz$.<|im_end|>
<|im_start|>assistant
<Z_47><Z_10><Z_39><ANSWER>

Question: Given: $x : y : z = 1 : 2 : 7$ and $2x - y + 3z = 105$, find the value of $xyz$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is at the candy store buying jellybeans. He plans to buy at least 150 jellybeans. He wants to buy the minimum amount such that he would have exactly $14$ leftover after dividing the jellybeans evenly among $15$ people. How many jellybeans should Alex buy?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is at the candy store buying jellybeans. He plans to buy at least 150 jellybeans. He wants to buy the minimum amount such that he would have exactly $14$ leftover after dividing the jellybeans evenly among $15$ people. How many jellybeans should Alex buy?<|im_end|>
<|im_start|>assistant
<Z_57><Z_82><Z_47><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is at the candy store buying jellybeans. He plans to buy at least 150 jellybeans. He wants to buy the minimum amount such that he would have exactly $14$ leftover after dividing the jellybeans evenly among $15$ people. How many jellybeans should Alex buy?<|im_end|>
<|im_start|>assistant
<Z_6><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is at the candy store buying jellybeans. He plans to buy at least 150 jellybeans. He wants to buy the minimum amount such that he would have exactly $14$ leftover after dividing the jellybeans evenly among $15$ people. How many jellybeans should Alex buy?<|im_end|>
<|im_start|>assistant
<Z_20><Z_69><Z_20><Z_87><ANSWER>

Question: Alex is at the candy store buying jellybeans. He plans to buy at least 150 jellybeans. He wants to buy the minimum amount such that he would have exactly $14$ leftover after dividing the jellybeans evenly among $15$ people. How many jellybeans should Alex buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = x^3$ and $f(x) = 3x^2 - 2x + 1$, find the value of $f(g(3))$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = x^3$ and $f(x) = 3x^2 - 2x + 1$, find the value of $f(g(3))$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_34><Z_31><Z_124><Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = x^3$ and $f(x) = 3x^2 - 2x + 1$, find the value of $f(g(3))$.<|im_end|>
<|im_start|>assistant
<Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = x^3$ and $f(x) = 3x^2 - 2x + 1$, find the value of $f(g(3))$.<|im_end|>
<|im_start|>assistant
<Z_13><Z_50><Z_91><Z_113><Z_99><ANSWER>

Question: If $g(x) = x^3$ and $f(x) = 3x^2 - 2x + 1$, find the value of $f(g(3))$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $(-64)^{4/3}$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $(-64)^{4/3}$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_97><Z_30><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $(-64)^{4/3}$.<|im_end|>
<|im_start|>assistant
<Z_70><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $(-64)^{4/3}$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_37><Z_113><Z_127><ANSWER>

Question: Evaluate $(-64)^{4/3}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Excluding stoppages, the speed of the bus is some kmph, and including stoppages, it is 36 kmph. The bus stops for 20 minutes per hour. What is the speed of the bus excluding stoppages?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Excluding stoppages, the speed of the bus is some kmph, and including stoppages, it is 36 kmph. The bus stops for 20 minutes per hour. What is the speed of the bus excluding stoppages?<|im_end|>
<|im_start|>assistant
<Z_104><Z_108><Z_99><Z_41><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Excluding stoppages, the speed of the bus is some kmph, and including stoppages, it is 36 kmph. The bus stops for 20 minutes per hour. What is the speed of the bus excluding stoppages?<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Excluding stoppages, the speed of the bus is some kmph, and including stoppages, it is 36 kmph. The bus stops for 20 minutes per hour. What is the speed of the bus excluding stoppages?<|im_end|>
<|im_start|>assistant
<Z_34><Z_122><Z_57><Z_39><Z_74><ANSWER>

Question: Excluding stoppages, the speed of the bus is some kmph, and including stoppages, it is 36 kmph. The bus stops for 20 minutes per hour. What is the speed of the bus excluding stoppages?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = x^2f'(1) - 3x$, find the value of $f'(2)$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = x^2f'(1) - 3x$, find the value of $f'(2)$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_97><Z_92><Z_126><Z_103><Z_34><Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = x^2f'(1) - 3x$, find the value of $f'(2)$.<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = x^2f'(1) - 3x$, find the value of $f'(2)$.<|im_end|>
<|im_start|>assistant
<Z_100><Z_77><Z_80><Z_123><Z_57><Z_12><Z_52><ANSWER>

Question: Given the function $f(x) = x^2f'(1) - 3x$, find the value of $f'(2)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?<|im_end|>
<|im_start|>assistant
<Z_104><Z_91><Z_96><Z_92><Z_95><Z_101><Z_125><Z_90><Z_58><Z_16><Z_38><Z_45><Z_67><Z_41><Z_75><Z_53><Z_26><Z_34><Z_124><Z_76><Z_47><Z_92><Z_76><Z_81><Z_96><Z_125><Z_19><Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?<|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
On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?<|im_end|>
<|im_start|>assistant
<Z_41><Z_3><Z_48><Z_29><Z_86><Z_36><Z_64><Z_107><Z_16><Z_30><Z_47><Z_28><Z_97><Z_17><Z_0><Z_12><Z_56><Z_107><Z_34><Z_124><Z_81><Z_11><Z_100><Z_116><Z_72><Z_81><Z_115><Z_66><ANSWER>

Question: On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and passes through the point $(-2, 4)$. Determine the expression for this line in the form $y = mx + b$ and calculate the value of $m + b$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and passes through the point $(-2, 4)$. Determine the expression for this line in the form $y = mx + b$ and calculate the value of $m + b$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_59><Z_34><Z_47><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and passes through the point $(-2, 4)$. Determine the expression for this line in the form $y = mx + b$ and calculate the value of $m + b$.<|im_end|>
<|im_start|>assistant
<Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and passes through the point $(-2, 4)$. Determine the expression for this line in the form $y = mx + b$ and calculate the value of $m + b$.<|im_end|>
<|im_start|>assistant
<Z_8><Z_8><Z_56><Z_44><Z_64><ANSWER>

Question: A line has a slope of $5$ and passes through the point $(-2, 4)$. Determine the expression for this line in the form $y = mx + b$ and calculate the value of $m + b$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^a, where a is the a-number of the result of his die roll, and plots 3^a on a number line as the point a. Finally, he repeats this entire process, this time plotting the result as the point b. What is the probability that the distance between a and b is greater than the value of b?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^a, where a is the a-number of the result of his die roll, and plots 3^a on a number line as the point a. Finally, he repeats this entire process, this time plotting the result as the point b. What is the probability that the distance between a and b is greater than the value of b?<|im_end|>
<|im_start|>assistant
<Z_94><Z_29><Z_44><Z_110><Z_66><Z_50><Z_98><Z_92><Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^a, where a is the a-number of the result of his die roll, and plots 3^a on a number line as the point a. Finally, he repeats this entire process, this time plotting the result as the point b. What is the probability that the distance between a and b is greater than the value of b?<|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
Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^a, where a is the a-number of the result of his die roll, and plots 3^a on a number line as the point a. Finally, he repeats this entire process, this time plotting the result as the point b. What is the probability that the distance between a and b is greater than the value of b?<|im_end|>
<|im_start|>assistant
<Z_117><Z_25><Z_113><Z_17><Z_15><Z_119><Z_20><Z_102><Z_93><ANSWER>

Question: Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^a, where a is the a-number of the result of his die roll, and plots 3^a on a number line as the point a. Finally, he repeats this entire process, this time plotting the result as the point b. What is the probability that the distance between a and b is greater than the value of b?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An infinite geometric series has a common ratio of \( \frac{1}{4} \) and a sum of 80. What is the first term of the series?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An infinite geometric series has a common ratio of \( \frac{1}{4} \) and a sum of 80. What is the first term of the series?<|im_end|>
<|im_start|>assistant
<Z_104><Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An infinite geometric series has a common ratio of \( \frac{1}{4} \) and a sum of 80. What is the first term of the series?<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An infinite geometric series has a common ratio of \( \frac{1}{4} \) and a sum of 80. What is the first term of the series?<|im_end|>
<|im_start|>assistant
<Z_61><Z_103><ANSWER>

Question: An infinite geometric series has a common ratio of \( \frac{1}{4} \) and a sum of 80. What is the first term of the series?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, let \[f_{n}(x) = f_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N$, the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, let \[f_{n}(x) = f_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N$, the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$<|im_end|>
<|im_start|>assistant
<Z_104><Z_68><Z_78><Z_125><Z_58><Z_38><Z_41><Z_47><Z_56><Z_95><Z_38><Z_52><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, let \[f_{n}(x) = f_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N$, the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, let \[f_{n}(x) = f_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N$, the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$<|im_end|>
<|im_start|>assistant
<Z_63><Z_39><Z_69><Z_75><Z_84><Z_39><Z_75><Z_10><Z_10><Z_90><Z_116><Z_51><Z_34><ANSWER>

Question: Let $f_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, let \[f_{n}(x) = f_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N$, the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya wrote the natural numbers 1, 2, ..., 20 on the board. Vasya first erased all even numbers and then erased all numbers that give a remainder of 4 when divided by 5. How many numbers are left on the board?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya wrote the natural numbers 1, 2, ..., 20 on the board. Vasya first erased all even numbers and then erased all numbers that give a remainder of 4 when divided by 5. How many numbers are left on the board?<|im_end|>
<|im_start|>assistant
<Z_104><Z_36><Z_34><Z_36><Z_92><Z_87><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya wrote the natural numbers 1, 2, ..., 20 on the board. Vasya first erased all even numbers and then erased all numbers that give a remainder of 4 when divided by 5. How many numbers are left on the board?<|im_end|>
<|im_start|>assistant
<Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Petya wrote the natural numbers 1, 2, ..., 20 on the board. Vasya first erased all even numbers and then erased all numbers that give a remainder of 4 when divided by 5. How many numbers are left on the board?<|im_end|>
<|im_start|>assistant
<Z_34><Z_59><Z_68><Z_96><Z_73><Z_69><Z_24><ANSWER>

Question: Petya wrote the natural numbers 1, 2, ..., 20 on the board. Vasya first erased all even numbers and then erased all numbers that give a remainder of 4 when divided by 5. How many numbers are left on the board?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying.<|im_end|>
<|im_start|>assistant
<Z_104><Z_107><Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying.<|im_end|>
<|im_start|>assistant
<Z_29><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying.<|im_end|>
<|im_start|>assistant
<Z_30><Z_19><Z_106><ANSWER>

Question:
10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the numerical value of $k$ for which
\[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the numerical value of $k$ for which
\[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]<|im_end|>
<|im_start|>assistant
<Z_57><Z_115><Z_41><Z_61><Z_31><Z_107><Z_70><Z_75><Z_20><Z_115><Z_126><Z_52><Z_60><Z_49><Z_124><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the numerical value of $k$ for which
\[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]<|im_end|>
<|im_start|>assistant
<Z_0><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the numerical value of $k$ for which
\[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]<|im_end|>
<|im_start|>assistant
<Z_80><Z_58><Z_125><Z_111><Z_69><Z_87><Z_28><Z_60><Z_24><Z_46><Z_106><Z_56><Z_63><Z_103><Z_66><Z_70><ANSWER>

Question:
Find the numerical value of $k$ for which
\[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Pete's memory card can hold 3,000 pictures of 8 megabytes each. How many pictures can it hold of 6 megabytes each?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Pete's memory card can hold 3,000 pictures of 8 megabytes each. How many pictures can it hold of 6 megabytes each?<|im_end|>
<|im_start|>assistant
<Z_57><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Pete's memory card can hold 3,000 pictures of 8 megabytes each. How many pictures can it hold of 6 megabytes each?<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Pete's memory card can hold 3,000 pictures of 8 megabytes each. How many pictures can it hold of 6 megabytes each?<|im_end|>
<|im_start|>assistant
<Z_103><Z_121><ANSWER>

Question: Pete's memory card can hold 3,000 pictures of 8 megabytes each. How many pictures can it hold of 6 megabytes each?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the areas of two squares with side lengths of 8 and 10 respectively.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the areas of two squares with side lengths of 8 and 10 respectively.<|im_end|>
<|im_start|>assistant
<Z_106><Z_63><Z_52><Z_82><Z_91><Z_90><Z_82><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the areas of two squares with side lengths of 8 and 10 respectively.<|im_end|>
<|im_start|>assistant
<Z_26><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the areas of two squares with side lengths of 8 and 10 respectively.<|im_end|>
<|im_start|>assistant
<Z_106><Z_115><Z_32><Z_19><Z_77><Z_84><Z_54><Z_94><ANSWER>

Question: Find the sum of the areas of two squares with side lengths of 8 and 10 respectively.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tracy set up a booth at an art fair. 20 people came to look at her art. Four of those customers bought two paintings each. The next 12 of those customers bought one painting each. The last 4 customers bought four paintings each.  How many paintings did Tracy sell at the art fair?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tracy set up a booth at an art fair. 20 people came to look at her art. Four of those customers bought two paintings each. The next 12 of those customers bought one painting each. The last 4 customers bought four paintings each.  How many paintings did Tracy sell at the art fair?<|im_end|>
<|im_start|>assistant
<Z_57><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tracy set up a booth at an art fair. 20 people came to look at her art. Four of those customers bought two paintings each. The next 12 of those customers bought one painting each. The last 4 customers bought four paintings each.  How many paintings did Tracy sell at the art fair?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tracy set up a booth at an art fair. 20 people came to look at her art. Four of those customers bought two paintings each. The next 12 of those customers bought one painting each. The last 4 customers bought four paintings each.  How many paintings did Tracy sell at the art fair?<|im_end|>
<|im_start|>assistant
<Z_63><Z_66><ANSWER>

Question: Tracy set up a booth at an art fair. 20 people came to look at her art. Four of those customers bought two paintings each. The next 12 of those customers bought one painting each. The last 4 customers bought four paintings each.  How many paintings did Tracy sell at the art fair?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An event has some adults and 20 children. The event planner wants to make each table identical, with the same combination of adults and children and no people left over. The greatest number of tables the planner can set up is 4. How many adults are at the event?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An event has some adults and 20 children. The event planner wants to make each table identical, with the same combination of adults and children and no people left over. The greatest number of tables the planner can set up is 4. How many adults are at the event?<|im_end|>
<|im_start|>assistant
<Z_94><Z_93><Z_50><Z_99><Z_65><Z_39><Z_76><Z_95><Z_103><Z_26><Z_19><Z_91><Z_39><Z_118><Z_61><Z_47><Z_80><Z_16><Z_90><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An event has some adults and 20 children. The event planner wants to make each table identical, with the same combination of adults and children and no people left over. The greatest number of tables the planner can set up is 4. How many adults are at the event?<|im_end|>
<|im_start|>assistant
<Z_112><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An event has some adults and 20 children. The event planner wants to make each table identical, with the same combination of adults and children and no people left over. The greatest number of tables the planner can set up is 4. How many adults are at the event?<|im_end|>
<|im_start|>assistant
<Z_8><Z_106><Z_84><Z_61><Z_39><Z_47><Z_85><Z_70><Z_31><Z_16><Z_100><Z_121><Z_92><Z_64><Z_98><Z_118><Z_51><Z_19><Z_83><Z_19><ANSWER>

Question: An event has some adults and 20 children. The event planner wants to make each table identical, with the same combination of adults and children and no people left over. The greatest number of tables the planner can set up is 4. How many adults are at the event?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \(\mathbf{Z}\) is the set of integers, and sets \(A\), \(B\), and \(C\) are defined as follows:

\[A = \left\{ x \mid | x - 3 | < \pi, x \in \mathbf{Z} \right\} \]

\[B = \left\{ x \mid x^2 - 11x + 5 < 0, x \in \mathbf{Z} \right\} \]

\[C = \left\{ x \mid |2x^2 - 11x + 10| \geq |3x - 2|, x \in \mathbf{Z} \right\} \]

Let \(\bar{C}\) be the complement of \(C\) in \(\mathbf{Z}\). The number of elements in \(A \cap B \cap \bar{C}\) is ( ).

A. 7

B. 8

C. 15

D. 16<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \(\mathbf{Z}\) is the set of integers, and sets \(A\), \(B\), and \(C\) are defined as follows:

\[A = \left\{ x \mid | x - 3 | < \pi, x \in \mathbf{Z} \right\} \]

\[B = \left\{ x \mid x^2 - 11x + 5 < 0, x \in \mathbf{Z} \right\} \]

\[C = \left\{ x \mid |2x^2 - 11x + 10| \geq |3x - 2|, x \in \mathbf{Z} \right\} \]

Let \(\bar{C}\) be the complement of \(C\) in \(\mathbf{Z}\). The number of elements in \(A \cap B \cap \bar{C}\) is ( ).

A. 7

B. 8

C. 15

D. 16<|im_end|>
<|im_start|>assistant
<Z_94><Z_106><Z_124><Z_78><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \(\mathbf{Z}\) is the set of integers, and sets \(A\), \(B\), and \(C\) are defined as follows:

\[A = \left\{ x \mid | x - 3 | < \pi, x \in \mathbf{Z} \right\} \]

\[B = \left\{ x \mid x^2 - 11x + 5 < 0, x \in \mathbf{Z} \right\} \]

\[C = \left\{ x \mid |2x^2 - 11x + 10| \geq |3x - 2|, x \in \mathbf{Z} \right\} \]

Let \(\bar{C}\) be the complement of \(C\) in \(\mathbf{Z}\). The number of elements in \(A \cap B \cap \bar{C}\) is ( ).

A. 7

B. 8

C. 15

D. 16<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \(\mathbf{Z}\) is the set of integers, and sets \(A\), \(B\), and \(C\) are defined as follows:

\[A = \left\{ x \mid | x - 3 | < \pi, x \in \mathbf{Z} \right\} \]

\[B = \left\{ x \mid x^2 - 11x + 5 < 0, x \in \mathbf{Z} \right\} \]

\[C = \left\{ x \mid |2x^2 - 11x + 10| \geq |3x - 2|, x \in \mathbf{Z} \right\} \]

Let \(\bar{C}\) be the complement of \(C\) in \(\mathbf{Z}\). The number of elements in \(A \cap B \cap \bar{C}\) is ( ).

A. 7

B. 8

C. 15

D. 16<|im_end|>
<|im_start|>assistant
<Z_97><Z_20><Z_22><Z_36><Z_35><ANSWER>

Question:
Given that \(\mathbf{Z}\) is the set of integers, and sets \(A\), \(B\), and \(C\) are defined as follows:

\[A = \left\{ x \mid | x - 3 | < \pi, x \in \mathbf{Z} \right\} \]

\[B = \left\{ x \mid x^2 - 11x + 5 < 0, x \in \mathbf{Z} \right\} \]

\[C = \left\{ x \mid |2x^2 - 11x + 10| \geq |3x - 2|, x \in \mathbf{Z} \right\} \]

Let \(\bar{C}\) be the complement of \(C\) in \(\mathbf{Z}\). The number of elements in \(A \cap B \cap \bar{C}\) is ( ).

A. 7

B. 8

C. 15

D. 16


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the first 5 multiples of a certain number is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the first 5 multiples of a certain number is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_94><Z_73><Z_110><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the first 5 multiples of a certain number is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the first 5 multiples of a certain number is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_28><Z_87><Z_34><Z_56><ANSWER>

Question: The average of the first 5 multiples of a certain number is 27. What is the number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of spinsters to cats is 2 to 7 . if there are 55 more cats than spinsters , how many spinsters are there ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of spinsters to cats is 2 to 7 . if there are 55 more cats than spinsters , how many spinsters are there ?<|im_end|>
<|im_start|>assistant
<Z_57><Z_120><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of spinsters to cats is 2 to 7 . if there are 55 more cats than spinsters , how many spinsters are there ?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of spinsters to cats is 2 to 7 . if there are 55 more cats than spinsters , how many spinsters are there ?<|im_end|>
<|im_start|>assistant
<Z_80><Z_6><ANSWER>

Question: the ratio of spinsters to cats is 2 to 7 . if there are 55 more cats than spinsters , how many spinsters are there ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if x ^ 2 + y ^ 2 = 15 and xy = 3 , then ( x − y ) ^ 2 =<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if x ^ 2 + y ^ 2 = 15 and xy = 3 , then ( x − y ) ^ 2 =<|im_end|>
<|im_start|>assistant
<Z_111><ANSWER>

Question: if x ^ 2 + y ^ 2 = 15 and xy = 3 , then ( x − y ) ^ 2 =


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals  $5$  red chips,  $3$  red tile are equal to  $ 8$  black tiles, and a black tile is worth  $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to  $560$  and there are no more than five tiles of the same color.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals  $5$  red chips,  $3$  red tile are equal to  $ 8$  black tiles, and a black tile is worth  $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to  $560$  and there are no more than five tiles of the same color.<|im_end|>
<|im_start|>assistant
<Z_104><Z_52><Z_88><Z_81><Z_31><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals  $5$  red chips,  $3$  red tile are equal to  $ 8$  black tiles, and a black tile is worth  $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to  $560$  and there are no more than five tiles of the same color.<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals  $5$  red chips,  $3$  red tile are equal to  $ 8$  black tiles, and a black tile is worth  $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to  $560$  and there are no more than five tiles of the same color.<|im_end|>
<|im_start|>assistant
<Z_93><Z_59><Z_8><Z_61><Z_93><ANSWER>

Question:
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals  $5$  red chips,  $3$  red tile are equal to  $ 8$  black tiles, and a black tile is worth  $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to  $560$  and there are no more than five tiles of the same color.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the difference between the largest number and the least number written with the digits 7, 3, 1, 4?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the difference between the largest number and the least number written with the digits 7, 3, 1, 4?<|im_end|>
<|im_start|>assistant
<Z_94><Z_87><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the difference between the largest number and the least number written with the digits 7, 3, 1, 4?<|im_end|>
<|im_start|>assistant
<Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the difference between the largest number and the least number written with the digits 7, 3, 1, 4?<|im_end|>
<|im_start|>assistant
<Z_16><Z_122><Z_116><ANSWER>

Question: What is the difference between the largest number and the least number written with the digits 7, 3, 1, 4?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cody had nine old video games he wanted to get rid of. He decided to give four of the games to his friend Jake, three games to his friend Sarah, and one game to his friend Luke. Each day from Monday to Friday he sold two games, but on Saturday he bought five new games. How many games does Cody have now?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cody had nine old video games he wanted to get rid of. He decided to give four of the games to his friend Jake, three games to his friend Sarah, and one game to his friend Luke. Each day from Monday to Friday he sold two games, but on Saturday he bought five new games. How many games does Cody have now?<|im_end|>
<|im_start|>assistant
<Z_57><Z_37><Z_73><Z_82><Z_120><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cody had nine old video games he wanted to get rid of. He decided to give four of the games to his friend Jake, three games to his friend Sarah, and one game to his friend Luke. Each day from Monday to Friday he sold two games, but on Saturday he bought five new games. How many games does Cody have now?<|im_end|>
<|im_start|>assistant
<Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cody had nine old video games he wanted to get rid of. He decided to give four of the games to his friend Jake, three games to his friend Sarah, and one game to his friend Luke. Each day from Monday to Friday he sold two games, but on Saturday he bought five new games. How many games does Cody have now?<|im_end|>
<|im_start|>assistant
<Z_54><Z_97><Z_125><Z_64><Z_5><ANSWER>

Question: Cody had nine old video games he wanted to get rid of. He decided to give four of the games to his friend Jake, three games to his friend Sarah, and one game to his friend Luke. Each day from Monday to Friday he sold two games, but on Saturday he bought five new games. How many games does Cody have now?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the function be $$f(x)=(x^{2}+ \frac {1}{x})^{n}$$, where $$n=5 \int_{ 0 }^{ \frac {\pi}{2} } \cos x dx$$, then the coefficient of $x^{4}$ in the expansion of $f(x)$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the function be $$f(x)=(x^{2}+ \frac {1}{x})^{n}$$, where $$n=5 \int_{ 0 }^{ \frac {\pi}{2} } \cos x dx$$, then the coefficient of $x^{4}$ in the expansion of $f(x)$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_113><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the function be $$f(x)=(x^{2}+ \frac {1}{x})^{n}$$, where $$n=5 \int_{ 0 }^{ \frac {\pi}{2} } \cos x dx$$, then the coefficient of $x^{4}$ in the expansion of $f(x)$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_25><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the function be $$f(x)=(x^{2}+ \frac {1}{x})^{n}$$, where $$n=5 \int_{ 0 }^{ \frac {\pi}{2} } \cos x dx$$, then the coefficient of $x^{4}$ in the expansion of $f(x)$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_122><Z_20><ANSWER>

Question: Let the function be $$f(x)=(x^{2}+ \frac {1}{x})^{n}$$, where $$n=5 \int_{ 0 }^{ \frac {\pi}{2} } \cos x dx$$, then the coefficient of $x^{4}$ in the expansion of $f(x)$ is \_\_\_\_\_\_.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = \sqrt{x-3}$, find the smallest real number $x$ such that $g(g(x))$ is defined when $g(x) \geq 5$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = \sqrt{x-3}$, find the smallest real number $x$ such that $g(g(x))$ is defined when $g(x) \geq 5$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_100><Z_41><Z_98><Z_92><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = \sqrt{x-3}$, find the smallest real number $x$ such that $g(g(x))$ is defined when $g(x) \geq 5$.<|im_end|>
<|im_start|>assistant
<Z_16><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $g(x) = \sqrt{x-3}$, find the smallest real number $x$ such that $g(g(x))$ is defined when $g(x) \geq 5$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_22><Z_122><Z_77><Z_113><Z_87><ANSWER>

Question: If $g(x) = \sqrt{x-3}$, find the smallest real number $x$ such that $g(g(x))$ is defined when $g(x) \geq 5$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?<|im_end|>
<|im_start|>assistant
<Z_29><Z_73><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?<|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
What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?<|im_end|>
<|im_start|>assistant
<Z_57><Z_126><Z_36><ANSWER>

Question: What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?<|im_end|>
<|im_start|>assistant
<Z_94><Z_81><Z_98><Z_48><Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?<|im_end|>
<|im_start|>assistant
<Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?<|im_end|>
<|im_start|>assistant
<Z_51><Z_87><Z_70><Z_85><Z_47><ANSWER>

Question: There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?


<|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_104><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_113><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_81><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_109><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_57><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_57><Z_93><Z_87><Z_108><Z_79><Z_65><Z_20><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_87><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_31><Z_107><Z_74><Z_6><Z_54><Z_17><Z_42><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_94><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_94><Z_34><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_81><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_78><Z_93><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_94><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_94><Z_37><Z_12><Z_66><Z_64><Z_82><Z_98><Z_95><Z_55><Z_93><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_101><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_10><Z_48><Z_10><Z_52><Z_76><Z_9><Z_20><Z_107><Z_61><Z_125><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_104><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_113><Z_73><Z_59><Z_120><Z_91><Z_103><Z_83><Z_8><Z_16><Z_126><Z_76><Z_54><Z_54><Z_60><Z_83><Z_53><Z_5><Z_75><Z_34><Z_72><Z_56><Z_37><Z_93><Z_56><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_71><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_107><Z_18><Z_106><Z_53><Z_34><Z_119><Z_41><Z_40><Z_54><Z_47><Z_95><Z_126><Z_48><Z_76><Z_77><Z_108><Z_67><Z_82><Z_86><Z_99><Z_118><Z_118><Z_88><Z_88><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_104><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_104><Z_107><Z_83><Z_99><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_68><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_121><Z_74><Z_91><Z_33><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_104><Z_29><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_57><Z_105><Z_65><Z_11><Z_37><Z_61><Z_83><Z_10><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_75><Z_98><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_19><Z_61><Z_8><Z_68><Z_99><Z_17><Z_71><Z_79><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_104><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_57><Z_37><Z_98><Z_87><Z_11><Z_96><Z_24><Z_11><Z_126><Z_75><Z_55><Z_92><Z_93><Z_83><Z_39><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_98><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_63><Z_83><Z_62><Z_33><Z_46><Z_3><Z_46><Z_99><Z_17><Z_31><Z_53><Z_48><Z_53><Z_47><Z_82><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_57><Z_27><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_113><Z_105><Z_30><Z_22><Z_72><Z_12><Z_95><Z_65><Z_32><Z_78><Z_76><Z_12><Z_103><Z_88><Z_32><Z_19><Z_47><Z_49><Z_101><Z_126><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_78><Z_75><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_69><Z_13><Z_124><Z_114><Z_36><Z_110><Z_94><Z_34><Z_54><Z_112><Z_12><Z_65><Z_5><Z_114><Z_18><Z_94><Z_36><Z_108><Z_51><Z_2><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_104><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_104><Z_86><Z_44><Z_59><Z_32><Z_59><Z_58><Z_63><Z_117><Z_117><Z_121><Z_39><Z_31><Z_58><Z_87><Z_63><Z_117><Z_22><Z_98><Z_56><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_54><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_11><Z_14><Z_50><Z_44><Z_100><Z_3><Z_106><Z_33><Z_87><Z_28><Z_32><Z_123><Z_16><Z_125><Z_118><Z_43><Z_50><Z_7><Z_54><Z_87><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_104><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_104><Z_33><Z_108><Z_119><Z_89><Z_126><Z_73><Z_121><Z_59><Z_101><Z_114><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_42><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_99><Z_11><Z_23><Z_110><Z_61><Z_72><Z_105><Z_47><Z_114><Z_14><Z_18><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_94><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_113><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_101><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_121><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_104><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_54><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_117><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_94><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_57><Z_64><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_123><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_102><Z_16><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_94><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_57><Z_64><Z_77><Z_49><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_106><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_14><Z_39><Z_63><Z_46><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_104><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_94><Z_77><Z_37><Z_119><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_67><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_99><Z_67><Z_91><Z_2><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_57><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_113><Z_100><Z_49><Z_98><Z_121><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_104><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_109><Z_61><Z_90><Z_87><Z_92><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_104><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_113><Z_49><Z_110><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_119><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_94><Z_12><Z_119><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_104><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_57><Z_115><Z_34><Z_126><Z_123><Z_95><Z_117><Z_91><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_107><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_13><Z_123><Z_7><Z_52><Z_127><Z_2><Z_79><Z_19><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_104><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_104><Z_40><Z_31><Z_81><Z_47><Z_5><Z_60><Z_82><Z_115><Z_80><Z_96><Z_66><Z_5><Z_91><Z_76><Z_101><Z_45><Z_117><Z_76><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_33><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_119><Z_4><Z_50><Z_118><Z_14><Z_52><Z_9><Z_78><Z_33><Z_52><Z_117><Z_82><Z_65><Z_97><Z_107><Z_15><Z_49><Z_117><Z_68><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_57><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_104><Z_43><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_125><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_30><Z_66><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?