<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|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
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|im_end|>
<|im_start|>assistant
<Z_57><Z_110><Z_120><Z_43><Z_40><Z_87><Z_16><Z_96><Z_47><Z_72><Z_81><Z_5><Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|im_end|>
<|im_start|>assistant
<Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|im_end|>
<|im_start|>assistant
<Z_18><Z_116><Z_62><Z_68><Z_12><Z_41><Z_83><Z_41><Z_2><Z_58><Z_64><Z_111><Z_19><ANSWER>

Question: Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|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 class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|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 of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|im_end|>
<|im_start|>assistant
<Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|im_end|>
<|im_start|>assistant
<Z_105><ANSWER>

Question: In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|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 ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|im_end|>
<|im_start|>assistant
<Z_57><Z_66><Z_97><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|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
the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|im_end|>
<|im_start|>assistant
<Z_85><Z_58><Z_79><ANSWER>

Question: the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_94><Z_40><Z_114><Z_82><Z_22><Z_43><Z_121><Z_87><Z_26><Z_59><Z_65><Z_33><Z_95><Z_99><Z_26><Z_83><Z_63><Z_45><Z_9><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_0><Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_73><Z_117><Z_2><Z_10><Z_113><Z_101><Z_50><Z_99><Z_120><Z_25><Z_26><Z_60><Z_94><Z_86><Z_11><Z_41><Z_53><Z_60><Z_1><Z_37><ANSWER>

Question: A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|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
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|im_end|>
<|im_start|>assistant
<Z_57><Z_51><Z_114><Z_108><Z_82><Z_114><Z_37><Z_126><Z_53><Z_115><Z_117><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|im_end|>
<|im_start|>assistant
<Z_30><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|im_end|>
<|im_start|>assistant
<Z_115><Z_105><Z_24><Z_123><Z_11><Z_49><Z_124><Z_123><Z_113><Z_22><Z_115><ANSWER>

Question: In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|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
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_104><Z_97><Z_34><Z_72><Z_87><Z_32><Z_87><Z_117><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_40><Z_94><Z_53><Z_118><Z_127><Z_3><Z_33><Z_96><ANSWER>

Question: For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|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
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|im_end|>
<|im_start|>assistant
<Z_94><Z_30><Z_97><Z_93><Z_79><Z_89><Z_75><Z_82><Z_123><Z_90><Z_96><Z_70><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|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
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|im_end|>
<|im_start|>assistant
<Z_73><Z_89><Z_127><Z_45><Z_8><Z_51><Z_9><Z_21><Z_47><Z_81><Z_57><Z_54><Z_27><ANSWER>

Question: Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|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
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_104><Z_83><Z_50><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_119><Z_83><Z_0><Z_115><ANSWER>

Question: In how many ways can you create a daily class schedule that includes 6 subjects out of 10?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|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
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|im_end|>
<|im_start|>assistant
<Z_57><Z_27><Z_4><Z_110><Z_107><Z_86><Z_119><Z_50><Z_10><Z_62><Z_37><Z_96><Z_124><Z_118><Z_61><Z_82><Z_41><Z_107><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|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
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|im_end|>
<|im_start|>assistant
<Z_51><Z_48><Z_59><Z_35><Z_51><Z_117><Z_6><Z_32><Z_61><Z_92><Z_29><Z_27><Z_6><Z_54><Z_42><Z_63><Z_125><Z_14><Z_60><ANSWER>

Question: Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|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
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|im_end|>
<|im_start|>assistant
<Z_94><Z_108><Z_50><Z_120><Z_92><Z_33><Z_66><Z_80><Z_72><Z_58><Z_107><Z_24><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|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
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|im_end|>
<|im_start|>assistant
<Z_51><Z_45><Z_4><Z_72><Z_69><Z_96><Z_32><Z_77><Z_64><Z_56><Z_48><Z_23><Z_73><ANSWER>

Question: Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|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 number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_104><Z_36><Z_97><Z_47><Z_87><Z_124><Z_49><Z_34><Z_66><Z_61><Z_87><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_61><Z_61><Z_96><Z_40><Z_103><Z_34><Z_85><Z_6><Z_77><Z_62><Z_85><Z_29><ANSWER>

Question: The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|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
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|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
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|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
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

Question: $a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|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 person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_113><Z_50><Z_37><Z_100><Z_49><Z_119><Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|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
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_91><Z_93><Z_47><Z_86><Z_81><Z_33><Z_88><ANSWER>

Question: A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|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
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|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
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

Question: Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|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 a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_106><Z_115><Z_70><Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_7><Z_47><Z_80><Z_117><ANSWER>

Question:
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|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
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|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
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|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
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

Question: Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|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
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_94><Z_65><Z_70><Z_107><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|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
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_40><Z_62><Z_49><Z_108><Z_64><ANSWER>

Question: tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|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
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_104><Z_13><Z_64><Z_65><Z_72><Z_78><Z_80><Z_81><Z_95><Z_41><Z_20><Z_123><Z_56><Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|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
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_90><Z_123><Z_68><Z_12><Z_105><Z_31><Z_6><Z_97><Z_78><Z_80><Z_76><Z_95><Z_73><Z_75><ANSWER>

Question:
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.<|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
Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

Question: Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 2010 for which the sum of the digits is 10?<|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 first year after 2010 for which the sum of the digits is 10?<|im_end|>
<|im_start|>assistant
<Z_57><Z_28><Z_92><Z_92><Z_115><Z_75><Z_78><Z_66><Z_123><Z_98><Z_64><Z_63><Z_96><Z_45><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 2010 for which the sum of the digits is 10?<|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
What is the first year after 2010 for which the sum of the digits is 10?<|im_end|>
<|im_start|>assistant
<Z_37><Z_67><Z_105><Z_59><Z_15><Z_122><Z_5><Z_1><Z_20><Z_31><Z_104><Z_124><Z_96><Z_56><Z_115><ANSWER>

Question: What is the first year after 2010 for which the sum of the digits is 10?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|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 person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_113><Z_115><Z_59><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_124><Z_24><Z_127><Z_13><ANSWER>

Question: A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|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 \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|im_end|>
<|im_start|>assistant
<Z_104><Z_37><Z_64><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|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
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|im_end|>
<|im_start|>assistant
<Z_37><Z_116><Z_122><Z_14><ANSWER>

Question: If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|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
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_94><Z_73><Z_32><Z_33><Z_33><Z_5><Z_91><Z_9><Z_5><Z_33><Z_66><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_7><Z_84><Z_28><Z_107><Z_17><Z_118><Z_70><Z_125><Z_76><Z_110><Z_53><Z_115><ANSWER>

Question: Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|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 boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_104><Z_97><Z_100><Z_4><Z_43><Z_100><Z_40><Z_100><Z_75><Z_65><Z_82><Z_80><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_105><Z_21><Z_110><Z_9><Z_13><Z_79><Z_82><Z_74><Z_71><Z_93><Z_84><Z_3><Z_4><ANSWER>

Question: A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|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 area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<Z_29><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<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 area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<Z_56><Z_22><ANSWER>

Question: What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|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 smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_57><Z_103><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|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
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_84><Z_34><Z_124><ANSWER>

Question: Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|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
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|im_end|>
<|im_start|>assistant
<Z_104><Z_70><Z_124><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|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
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|im_end|>
<|im_start|>assistant
<Z_122><Z_71><Z_33><Z_103><ANSWER>

Question:
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|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
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_94><Z_41><Z_99><Z_72><Z_91><Z_72><Z_32><Z_65><Z_63><Z_70><Z_118><Z_125><Z_80><Z_66><Z_87><Z_33><Z_95><Z_55><Z_47><Z_67><Z_22><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|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
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_94><Z_117><Z_81><Z_1><Z_41><Z_26><Z_96><Z_3><Z_113><Z_19><Z_38><Z_109><Z_115><Z_13><Z_90><Z_16><Z_14><Z_6><Z_103><Z_47><Z_58><Z_49><ANSWER>

Question: Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|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
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_113><Z_73><Z_80><Z_70><Z_20><Z_119><Z_124><Z_22><Z_34><Z_90><Z_119><Z_79><Z_49><Z_96><Z_99><Z_90><Z_52><Z_37><Z_10><Z_118><Z_64><Z_101><Z_11><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_24><Z_71><Z_13><Z_11><Z_3><Z_68><Z_37><Z_33><Z_98><Z_35><Z_105><Z_29><Z_124><Z_99><Z_30><Z_120><Z_16><Z_107><Z_75><Z_14><Z_62><Z_21><Z_1><ANSWER>

Question: Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|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
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_2><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_6><Z_87><ANSWER>

Question: Find the smallest positive integer whose cube ends in $392$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|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
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_94><Z_65><Z_110><Z_32><Z_99><Z_121><Z_119><Z_91><Z_124><Z_114><Z_124><Z_67><Z_83><Z_41><Z_93><Z_124><Z_52><Z_79><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_14><Z_42><Z_100><Z_77><Z_68><Z_68><Z_38><Z_29><Z_26><Z_11><Z_117><Z_127><Z_36><Z_57><Z_58><Z_111><Z_111><Z_99><Z_88><ANSWER>

Question: What is the greatest two-digit whole number, the product of whose digits is 12?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|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 cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|im_end|>
<|im_start|>assistant
<Z_57><Z_38><Z_64><Z_99><Z_49><Z_34><Z_96><Z_61><Z_9><Z_33><Z_48><Z_76><Z_75><Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|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
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|im_end|>
<|im_start|>assistant
<Z_124><Z_4><Z_37><Z_21><Z_17><Z_3><Z_32><Z_100><Z_14><Z_48><Z_68><Z_64><Z_36><Z_55><ANSWER>

Question: the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|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 man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_104><Z_105><Z_4><Z_103><Z_65><Z_45><Z_79><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_54><Z_93><Z_117><Z_58><Z_53><Z_125><Z_45><Z_90><ANSWER>

Question: A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|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
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_94><Z_13><Z_62><Z_62><Z_123><Z_63><Z_65><Z_107><Z_56><Z_41><Z_65><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_31><Z_39><Z_60><Z_84><Z_67><Z_15><Z_90><Z_62><Z_18><Z_22><Z_123><Z_105><ANSWER>

Question: Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|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 notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_104><Z_59><Z_93><Z_11><Z_96><Z_60><Z_76><Z_70><Z_80><Z_117><Z_49><Z_114><Z_98><Z_61><Z_48><Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_105><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_25><Z_106><Z_3><Z_107><Z_29><Z_107><Z_72><Z_48><Z_122><Z_86><Z_31><Z_37><Z_113><Z_36><Z_44><Z_125><ANSWER>

Question:
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|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 score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_94><Z_120><Z_83><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_109><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_112><Z_54><Z_11><Z_26><ANSWER>

Question: The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.<|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
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.<|im_end|>
<|im_start|>assistant
<Z_42><ANSWER>

Question: Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit whole numbers have no 5's and no 8's as digits?<|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 whole numbers have no 5's and no 8's as digits?<|im_end|>
<|im_start|>assistant
<Z_113><Z_86><Z_52><Z_103><Z_12><Z_56><Z_115><Z_12><Z_70><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 whole numbers have no 5's and no 8's as digits?<|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
How many three-digit whole numbers have no 5's and no 8's as digits?<|im_end|>
<|im_start|>assistant
<Z_39><Z_60><Z_90><Z_50><Z_100><Z_104><Z_83><Z_35><Z_37><ANSWER>

Question: How many three-digit whole numbers have no 5's and no 8's as digits?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|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 wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_104><Z_68><Z_27><Z_125><Z_4><Z_79><Z_59><Z_96><Z_96><Z_72><Z_39><Z_124><Z_26><Z_107><Z_126><Z_63><Z_119><Z_6><Z_37><Z_53><Z_52><Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_95><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_44><Z_98><Z_76><Z_68><Z_104><Z_13><Z_108><Z_21><Z_90><Z_85><Z_123><Z_122><Z_68><Z_68><Z_117><Z_48><Z_92><Z_106><Z_102><Z_58><Z_2><Z_77><ANSWER>

Question: A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_104><Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_94><Z_73><Z_99><Z_103><Z_41><Z_117><Z_88><Z_28><Z_39><Z_28><Z_22><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_1><Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_79><Z_16><Z_76><Z_85><Z_3><Z_126><Z_110><Z_127><Z_100><Z_90><Z_56><Z_123><ANSWER>

Question: Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|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
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_57><Z_27><Z_114><Z_66><Z_33><Z_92><Z_83><Z_88><Z_59><Z_93><Z_66><Z_118><Z_48><Z_125><Z_66><Z_39><Z_96><Z_55><Z_121><Z_92><Z_39><Z_47><Z_99><Z_66><Z_9><Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|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
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_77><Z_104><Z_97><Z_100><Z_123><Z_40><Z_86><Z_50><Z_28><Z_80><Z_71><Z_26><Z_10><Z_78><Z_59><Z_68><Z_25><Z_42><Z_49><Z_78><Z_30><Z_8><Z_118><Z_38><Z_80><Z_98><ANSWER>

Question: Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|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 jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_94><Z_106><Z_100><Z_65><Z_12><Z_22><Z_70><Z_80><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_90><Z_26><Z_118><Z_1><Z_77><Z_17><Z_74><Z_18><Z_15><ANSWER>

Question: A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|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
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|im_end|>
<|im_start|>assistant
<Z_94><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|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
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|im_end|>
<|im_start|>assistant
<Z_10><Z_34><ANSWER>

Question: Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|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 picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|im_end|>
<|im_start|>assistant
<Z_94><Z_47><Z_41><Z_108><Z_70><Z_88><Z_61><Z_93><Z_22><Z_54><Z_72><Z_16><Z_38><Z_37><Z_98><Z_95><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|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
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|im_end|>
<|im_start|>assistant
<Z_65><Z_63><Z_22><Z_31><Z_101><Z_62><Z_70><Z_55><Z_121><Z_32><Z_89><Z_44><Z_105><Z_31><Z_61><Z_111><Z_113><ANSWER>

Question:
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|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 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_104><Z_64><Z_77><Z_64><Z_12><Z_100><Z_26><Z_49><Z_99><Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_25><Z_47><Z_73><Z_12><Z_2><Z_26><Z_103><Z_10><Z_113><Z_80><ANSWER>

Question: How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|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
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_43><Z_87><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|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
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|im_end|>
<|im_start|>assistant
<Z_51><Z_74><Z_6><Z_75><ANSWER>

Question: Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|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
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_94><Z_51><Z_34><Z_41><Z_87><Z_75><Z_41><Z_55><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_76><Z_115><Z_92><Z_55><Z_51><Z_8><Z_27><Z_124><ANSWER>

Question: Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|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
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_104><Z_1><Z_110><Z_41><Z_47><Z_10><Z_58><Z_92><Z_124><Z_82><Z_45><Z_82><Z_37><Z_45><Z_70><Z_126><Z_121><Z_126><Z_91><Z_54><Z_70><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|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
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_38><Z_25><Z_97><Z_63><Z_73><Z_100><Z_12><Z_97><Z_91><Z_79><Z_101><Z_14><Z_39><Z_104><Z_18><Z_86><Z_77><Z_80><Z_19><Z_60><Z_120><ANSWER>

Question: Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|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
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_104><Z_1><Z_124><Z_90><Z_28><Z_38><Z_53><Z_26><Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_109><Z_62><Z_79><Z_53><Z_60><Z_65><Z_71><Z_88><Z_96><ANSWER>

Question:
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?


<|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 rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|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 sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

Question: Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|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
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|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 units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

Question: Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|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 can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_94><Z_110><Z_34><Z_100><Z_81><Z_71><Z_91><Z_19><Z_96><Z_87><Z_61><Z_52><Z_33><Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_18><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_53><Z_12><Z_71><Z_2><Z_53><Z_32><Z_126><Z_77><Z_12><Z_101><Z_57><Z_28><Z_4><Z_114><ANSWER>

Question: A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|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
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_57><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_99><Z_63><ANSWER>

Question:
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|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 constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_86><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_114><Z_105><Z_75><ANSWER>

Question: Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|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 product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|im_end|>
<|im_start|>assistant
<Z_113><Z_89><Z_37><Z_36><Z_65><Z_88><Z_49><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|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
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|im_end|>
<|im_start|>assistant
<Z_15><Z_83><Z_48><Z_24><Z_86><Z_64><Z_93><Z_87><ANSWER>

Question: If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?