<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|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
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|im_end|>
<|im_start|>assistant
<Z_104><Z_125><Z_88><Z_92><Z_125><Z_66><Z_55><Z_33><Z_63><Z_93><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|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
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|im_end|>
<|im_start|>assistant
<Z_70><Z_32><Z_75><Z_42><Z_0><Z_102><Z_89><Z_46><Z_69><Z_71><Z_67><ANSWER>

Question:
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|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
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_104><Z_62><Z_62><Z_120><Z_115><Z_119><Z_93><Z_82><Z_91><Z_33><Z_83><Z_90><Z_123><Z_26><Z_52><Z_118><Z_41><Z_32><Z_117><Z_88><Z_124><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_74><Z_122><Z_124><Z_127><Z_28><Z_2><Z_118><Z_105><Z_34><Z_81><Z_116><Z_33><Z_80><Z_56><Z_111><Z_107><Z_88><Z_79><Z_8><Z_70><Z_5><Z_2><ANSWER>

Question: If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|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 person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|im_end|>
<|im_start|>assistant
<Z_94><Z_27><Z_51><Z_89><Z_97><Z_86><Z_54><Z_65><Z_70><Z_91><Z_58><Z_56><Z_114><Z_75><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|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 person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|im_end|>
<|im_start|>assistant
<Z_21><Z_89><Z_47><Z_82><Z_25><Z_109><Z_42><Z_69><Z_53><Z_81><Z_122><Z_118><Z_109><Z_52><Z_105><ANSWER>

Question: A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|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
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|im_end|>
<|im_start|>assistant
<Z_94><Z_34><Z_108><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|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
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|im_end|>
<|im_start|>assistant
<Z_90><Z_96><Z_104><Z_22><ANSWER>

Question: A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|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
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_62><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_64><Z_85><ANSWER>

Question:
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|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 movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|im_end|>
<|im_start|>assistant
<Z_57><Z_50><Z_78><Z_83><Z_50><Z_59><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|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 movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|im_end|>
<|im_start|>assistant
<Z_13><Z_101><Z_74><Z_100><Z_13><Z_94><Z_22><ANSWER>

Question: A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|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
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_57><Z_86><Z_4><Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_71><Z_16><Z_20><Z_50><ANSWER>

Question: If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|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 candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|im_end|>
<|im_start|>assistant
<Z_74><Z_38><Z_120><Z_49><Z_51><Z_108><Z_73><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|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
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|im_end|>
<|im_start|>assistant
<Z_89><Z_105><Z_115><Z_120><Z_1><Z_17><Z_57><Z_107><ANSWER>

Question: A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_51><Z_50><Z_91><Z_51><Z_48><Z_114><Z_82><Z_75><Z_98><Z_115><Z_107><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|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
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_10><Z_16><Z_107><Z_34><Z_117><Z_86><Z_12><Z_84><Z_6><Z_34><Z_31><Z_19><Z_10><ANSWER>

Question: Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|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 train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_57><Z_34><Z_115><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_82><Z_26><Z_126><Z_116><ANSWER>

Question: A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|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
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_94><Z_27><Z_107><Z_89><Z_108><Z_119><Z_107><Z_47><Z_118><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_7><Z_113><Z_63><Z_126><Z_43><Z_28><Z_76><Z_65><Z_43><Z_71><ANSWER>

Question: On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|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 ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_57><Z_78><Z_81><Z_37><Z_59><Z_71><Z_26><Z_117><Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_112><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_35><Z_34><Z_3><Z_52><Z_22><Z_33><Z_13><Z_10><Z_83><ANSWER>

Question: How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|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
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_113><Z_106><Z_114><Z_58><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_74><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_86><Z_9><Z_41><Z_80><Z_79><ANSWER>

Question: James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|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
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|im_end|>
<|im_start|>assistant
<Z_57><Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|im_end|>
<|im_start|>assistant
<Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|im_end|>
<|im_start|>assistant
<Z_5><Z_14><ANSWER>

Question: Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|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 coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<Z_57><Z_115><Z_87><Z_108><Z_62><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<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 coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<Z_96><Z_119><Z_127><Z_14><Z_54><Z_45><ANSWER>

Question: What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|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
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|im_end|>
<|im_start|>assistant
<Z_57><Z_93><Z_48><Z_52><Z_63><Z_121><Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|im_end|>
<|im_start|>assistant
<Z_42><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|im_end|>
<|im_start|>assistant
<Z_39><Z_52><Z_28><Z_123><Z_15><Z_79><Z_15><ANSWER>

Question: To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John bakes 12 coconut macaroons, each weighing 5 ounces. He then packs an equal number of the macaroons in 4 different brown bags, ready for delivery. When he briefly leaves the kitchen to pick the phone, his little brother Steve eats the entire contents of a certain number of the brown bags. The total weight of the remaining coconut macaroons is 45 ounces. How many brown bags did Steve eat the contents of?<|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
John bakes 12 coconut macaroons, each weighing 5 ounces. He then packs an equal number of the macaroons in 4 different brown bags, ready for delivery. When he briefly leaves the kitchen to pick the phone, his little brother Steve eats the entire contents of a certain number of the brown bags. The total weight of the remaining coconut macaroons is 45 ounces. How many brown bags did Steve eat the contents of?<|im_end|>
<|im_start|>assistant
<Z_106><Z_32><Z_93><Z_80><Z_79><Z_11><Z_126><Z_72><Z_78><Z_118><Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John bakes 12 coconut macaroons, each weighing 5 ounces. He then packs an equal number of the macaroons in 4 different brown bags, ready for delivery. When he briefly leaves the kitchen to pick the phone, his little brother Steve eats the entire contents of a certain number of the brown bags. The total weight of the remaining coconut macaroons is 45 ounces. How many brown bags did Steve eat the contents of?<|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
John bakes 12 coconut macaroons, each weighing 5 ounces. He then packs an equal number of the macaroons in 4 different brown bags, ready for delivery. When he briefly leaves the kitchen to pick the phone, his little brother Steve eats the entire contents of a certain number of the brown bags. The total weight of the remaining coconut macaroons is 45 ounces. How many brown bags did Steve eat the contents of?<|im_end|>
<|im_start|>assistant
<Z_22><Z_127><Z_105><Z_82><Z_17><Z_79><Z_96><Z_16><Z_41><Z_86><Z_105><ANSWER>

Question: John bakes 12 coconut macaroons, each weighing 5 ounces. He then packs an equal number of the macaroons in 4 different brown bags, ready for delivery. When he briefly leaves the kitchen to pick the phone, his little brother Steve eats the entire contents of a certain number of the brown bags. The total weight of the remaining coconut macaroons is 45 ounces. How many brown bags did Steve eat the contents of?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man swims downstream 40 km and upstream 30 km taking 5 hours each time , what is the speed of the man in still water ?<|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 man swims downstream 40 km and upstream 30 km taking 5 hours each time , what is the speed of the man in still water ?<|im_end|>
<|im_start|>assistant
<Z_57><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man swims downstream 40 km and upstream 30 km taking 5 hours each time , what is the speed of the man in still water ?<|im_end|>
<|im_start|>assistant
<Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a man swims downstream 40 km and upstream 30 km taking 5 hours each time , what is the speed of the man in still water ?<|im_end|>
<|im_start|>assistant
<Z_97><Z_81><ANSWER>

Question: a man swims downstream 40 km and upstream 30 km taking 5 hours each time , what is the speed of the man in still water ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a tree of height 36 m is on one edge of a road broke at a certain height . it fell in such a way that the top of the tree touches the other edge of the road . if the breadth of the road is 12 m , then what is the height at which the tree broke ?<|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 tree of height 36 m is on one edge of a road broke at a certain height . it fell in such a way that the top of the tree touches the other edge of the road . if the breadth of the road is 12 m , then what is the height at which the tree broke ?<|im_end|>
<|im_start|>assistant
<Z_94><Z_30><Z_4><Z_124><Z_16><Z_90><Z_37><Z_26><Z_16><Z_37><Z_33><Z_39><Z_76><Z_95><Z_118><Z_72><Z_101><Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a tree of height 36 m is on one edge of a road broke at a certain height . it fell in such a way that the top of the tree touches the other edge of the road . if the breadth of the road is 12 m , then what is the height at which the tree broke ?<|im_end|>
<|im_start|>assistant
<Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a tree of height 36 m is on one edge of a road broke at a certain height . it fell in such a way that the top of the tree touches the other edge of the road . if the breadth of the road is 12 m , then what is the height at which the tree broke ?<|im_end|>
<|im_start|>assistant
<Z_121><Z_73><Z_27><Z_46><Z_3><Z_41><Z_115><Z_104><Z_49><Z_43><Z_79><Z_48><Z_18><Z_44><Z_21><Z_85><Z_73><Z_75><ANSWER>

Question: a tree of height 36 m is on one edge of a road broke at a certain height . it fell in such a way that the top of the tree touches the other edge of the road . if the breadth of the road is 12 m , then what is the height at which the tree broke ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Thomas made some stacks of wooden blocks. The first stack was 7 blocks tall. The second stack was 3 blocks taller than the first. The third stack was 6 blocks shorter than the second stack, and the fourth stack was 10 blocks taller than the third stack. The fifth stack has twice as many blocks as the second stack. How many blocks did Thomas use in all?<|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
Thomas made some stacks of wooden blocks. The first stack was 7 blocks tall. The second stack was 3 blocks taller than the first. The third stack was 6 blocks shorter than the second stack, and the fourth stack was 10 blocks taller than the third stack. The fifth stack has twice as many blocks as the second stack. How many blocks did Thomas use in all?<|im_end|>
<|im_start|>assistant
<Z_57><Z_89><Z_12><Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Thomas made some stacks of wooden blocks. The first stack was 7 blocks tall. The second stack was 3 blocks taller than the first. The third stack was 6 blocks shorter than the second stack, and the fourth stack was 10 blocks taller than the third stack. The fifth stack has twice as many blocks as the second stack. How many blocks did Thomas use in all?<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Thomas made some stacks of wooden blocks. The first stack was 7 blocks tall. The second stack was 3 blocks taller than the first. The third stack was 6 blocks shorter than the second stack, and the fourth stack was 10 blocks taller than the third stack. The fifth stack has twice as many blocks as the second stack. How many blocks did Thomas use in all?<|im_end|>
<|im_start|>assistant
<Z_87><Z_58><Z_41><Z_24><ANSWER>

Question: Thomas made some stacks of wooden blocks. The first stack was 7 blocks tall. The second stack was 3 blocks taller than the first. The third stack was 6 blocks shorter than the second stack, and the fourth stack was 10 blocks taller than the third stack. The fifth stack has twice as many blocks as the second stack. How many blocks did Thomas use in all?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Josh had 7 marbles in his collection. He lost 8 marbles and found 10 new ones.. How many more marbles did he find than those he lost?<|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
Josh had 7 marbles in his collection. He lost 8 marbles and found 10 new ones.. How many more marbles did he find than those he lost?<|im_end|>
<|im_start|>assistant
<Z_94><Z_36><Z_82><Z_36><Z_48><Z_80><Z_31><Z_26><Z_10><Z_64><Z_53><Z_101><Z_92><Z_95><Z_93><Z_58><Z_87><Z_47><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Josh had 7 marbles in his collection. He lost 8 marbles and found 10 new ones.. How many more marbles did he find than those he lost?<|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
Josh had 7 marbles in his collection. He lost 8 marbles and found 10 new ones.. How many more marbles did he find than those he lost?<|im_end|>
<|im_start|>assistant
<Z_99><Z_53><Z_90><Z_58><Z_86><Z_83><Z_3><Z_60><Z_30><Z_5><Z_86><Z_42><Z_26><Z_80><Z_28><Z_125><Z_125><Z_36><Z_101><ANSWER>

Question: Josh had 7 marbles in his collection. He lost 8 marbles and found 10 new ones.. How many more marbles did he find than those he lost?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two trains of equal are running on parallel lines in the same direction at 44 km / hr and 36 km / hr . the faster train passes the slower train in 36 sec . the length of each train is ?<|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 trains of equal are running on parallel lines in the same direction at 44 km / hr and 36 km / hr . the faster train passes the slower train in 36 sec . the length of each train is ?<|im_end|>
<|im_start|>assistant
<Z_94><Z_125><Z_13><Z_81><Z_99><Z_40><Z_16><Z_114><Z_114><Z_54><Z_90><Z_82><Z_11><Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two trains of equal are running on parallel lines in the same direction at 44 km / hr and 36 km / hr . the faster train passes the slower train in 36 sec . the length of each train is ?<|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
two trains of equal are running on parallel lines in the same direction at 44 km / hr and 36 km / hr . the faster train passes the slower train in 36 sec . the length of each train is ?<|im_end|>
<|im_start|>assistant
<Z_103><Z_21><Z_44><Z_92><Z_28><Z_88><Z_94><Z_85><Z_11><Z_69><Z_57><Z_104><Z_63><Z_124><ANSWER>

Question: two trains of equal are running on parallel lines in the same direction at 44 km / hr and 36 km / hr . the faster train passes the slower train in 36 sec . the length of each train is ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even four-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$?<|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 even four-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even four-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$?<|im_end|>
<|im_start|>assistant
<Z_127><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even four-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$?<|im_end|>
<|im_start|>assistant
<Z_29><Z_99><ANSWER>

Question: How many even four-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x < m$. To graph a congruence, we mark every point $(x, y)$ that satisfies the congruence. Consider the graphs of the following equations on modulo $9$ graph paper: $$y \equiv 2x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$
Determine the sum of the $x$-coordinates of the points where the graphs of the two equations intersect.<|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
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x < m$. To graph a congruence, we mark every point $(x, y)$ that satisfies the congruence. Consider the graphs of the following equations on modulo $9$ graph paper: $$y \equiv 2x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$
Determine the sum of the $x$-coordinates of the points where the graphs of the two equations intersect.<|im_end|>
<|im_start|>assistant
<Z_94><Z_49><Z_37><Z_30><Z_83><Z_33><Z_76><Z_60><Z_59><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x < m$. To graph a congruence, we mark every point $(x, y)$ that satisfies the congruence. Consider the graphs of the following equations on modulo $9$ graph paper: $$y \equiv 2x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$
Determine the sum of the $x$-coordinates of the points where the graphs of the two equations intersect.<|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
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x < m$. To graph a congruence, we mark every point $(x, y)$ that satisfies the congruence. Consider the graphs of the following equations on modulo $9$ graph paper: $$y \equiv 2x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$
Determine the sum of the $x$-coordinates of the points where the graphs of the two equations intersect.<|im_end|>
<|im_start|>assistant
<Z_45><Z_121><Z_85><Z_94><Z_94><Z_106><Z_113><Z_3><Z_38><Z_11><ANSWER>

Question:
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x < m$. To graph a congruence, we mark every point $(x, y)$ that satisfies the congruence. Consider the graphs of the following equations on modulo $9$ graph paper: $$y \equiv 2x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$
Determine the sum of the $x$-coordinates of the points where the graphs of the two equations intersect.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain company, there are some female managers. If 2/5 of all the employees are managers and 2/5 of all male employees are managers, the total number of female employees in the company is 750. How many female managers are there in the company?<|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 a certain company, there are some female managers. If 2/5 of all the employees are managers and 2/5 of all male employees are managers, the total number of female employees in the company is 750. How many female managers are there in the company?<|im_end|>
<|im_start|>assistant
<Z_113><Z_99><Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain company, there are some female managers. If 2/5 of all the employees are managers and 2/5 of all male employees are managers, the total number of female employees in the company is 750. How many female managers are there in the company?<|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
In a certain company, there are some female managers. If 2/5 of all the employees are managers and 2/5 of all male employees are managers, the total number of female employees in the company is 750. How many female managers are there in the company?<|im_end|>
<|im_start|>assistant
<Z_11><Z_110><Z_13><ANSWER>

Question: In a certain company, there are some female managers. If 2/5 of all the employees are managers and 2/5 of all male employees are managers, the total number of female employees in the company is 750. How many female managers are there in the company?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(1+i) = 0$, find the value of $a + b + c + d$.<|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
If the polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(1+i) = 0$, find the value of $a + b + c + d$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_40><Z_34><Z_103><Z_53><Z_114><Z_99><Z_12><Z_32><Z_39><Z_37><Z_72><Z_71><Z_34><Z_38><Z_38><Z_126><Z_90><Z_48><Z_12><Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(1+i) = 0$, find the value of $a + b + c + d$.<|im_end|>
<|im_start|>assistant
<Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(1+i) = 0$, find the value of $a + b + c + d$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_120><Z_99><Z_62><Z_39><Z_98><Z_42><Z_54><Z_26><Z_2><Z_26><Z_3><Z_11><Z_48><Z_80><Z_105><Z_105><Z_5><Z_43><Z_23><Z_105><ANSWER>

Question: If the polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(1+i) = 0$, find the value of $a + b + c + d$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f\left(\frac{1}{x}+1\right)=\frac{1}{{x^2}}+\frac{1}{x}+\frac{1}{4}$, find the value of $f\left(\frac{3}{2}\right)$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f\left(\frac{1}{x}+1\right)=\frac{1}{{x^2}}+\frac{1}{x}+\frac{1}{4}$, find the value of $f\left(\frac{3}{2}\right)$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_27><Z_87><Z_68><Z_40><Z_68><Z_70><Z_80><Z_40><Z_65><Z_76><Z_99><Z_31><Z_80><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f\left(\frac{1}{x}+1\right)=\frac{1}{{x^2}}+\frac{1}{x}+\frac{1}{4}$, find the value of $f\left(\frac{3}{2}\right)$.<|im_end|>
<|im_start|>assistant
<Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f\left(\frac{1}{x}+1\right)=\frac{1}{{x^2}}+\frac{1}{x}+\frac{1}{4}$, find the value of $f\left(\frac{3}{2}\right)$.<|im_end|>
<|im_start|>assistant
<Z_100><Z_62><Z_2><Z_122><Z_54><Z_95><Z_14><Z_73><Z_67><Z_89><Z_117><Z_105><Z_22><Z_69><Z_4><ANSWER>

Question: Given the function $f\left(\frac{1}{x}+1\right)=\frac{1}{{x^2}}+\frac{1}{x}+\frac{1}{4}$, find the value of $f\left(\frac{3}{2}\right)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A collection of 8 cubes consists of one cube with edge-length  $k$  for each integer  $k,\thinspace 1 \le k \le 8.$   A tower is to be built using all 8 cubes according to the rules:

 $\bullet$  Any cube may be the bottom cube in the tower.
 $\bullet$  The cube immediately on top of a cube with edge-length  $k$  must have edge-length at most  $k+2.$

Let  $T$  be the number of different towers than can be constructed.  What is the remainder when  $T$  is divided by 1000?<|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
A collection of 8 cubes consists of one cube with edge-length  $k$  for each integer  $k,\thinspace 1 \le k \le 8.$   A tower is to be built using all 8 cubes according to the rules:

 $\bullet$  Any cube may be the bottom cube in the tower.
 $\bullet$  The cube immediately on top of a cube with edge-length  $k$  must have edge-length at most  $k+2.$

Let  $T$  be the number of different towers than can be constructed.  What is the remainder when  $T$  is divided by 1000?<|im_end|>
<|im_start|>assistant
<Z_106><Z_27><Z_29><Z_4><Z_51><Z_115><Z_38><Z_82><Z_41><Z_120><Z_120><Z_114><Z_88><Z_65><Z_125><Z_89><Z_28><Z_125><Z_41><Z_52><Z_124><Z_100><Z_91><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A collection of 8 cubes consists of one cube with edge-length  $k$  for each integer  $k,\thinspace 1 \le k \le 8.$   A tower is to be built using all 8 cubes according to the rules:

 $\bullet$  Any cube may be the bottom cube in the tower.
 $\bullet$  The cube immediately on top of a cube with edge-length  $k$  must have edge-length at most  $k+2.$

Let  $T$  be the number of different towers than can be constructed.  What is the remainder when  $T$  is divided by 1000?<|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
A collection of 8 cubes consists of one cube with edge-length  $k$  for each integer  $k,\thinspace 1 \le k \le 8.$   A tower is to be built using all 8 cubes according to the rules:

 $\bullet$  Any cube may be the bottom cube in the tower.
 $\bullet$  The cube immediately on top of a cube with edge-length  $k$  must have edge-length at most  $k+2.$

Let  $T$  be the number of different towers than can be constructed.  What is the remainder when  $T$  is divided by 1000?<|im_end|>
<|im_start|>assistant
<Z_24><Z_67><Z_96><Z_1><Z_126><Z_126><Z_6><Z_77><Z_7><Z_62><Z_85><Z_72><Z_13><Z_97><Z_118><Z_16><Z_121><Z_50><Z_6><Z_53><Z_98><Z_95><Z_81><Z_112><ANSWER>

Question:
A collection of 8 cubes consists of one cube with edge-length  $k$  for each integer  $k,\thinspace 1 \le k \le 8.$   A tower is to be built using all 8 cubes according to the rules:

 $\bullet$  Any cube may be the bottom cube in the tower.
 $\bullet$  The cube immediately on top of a cube with edge-length  $k$  must have edge-length at most  $k+2.$

Let  $T$  be the number of different towers than can be constructed.  What is the remainder when  $T$  is divided by 1000?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some red peaches, 90 yellow peaches, and 16 green peaches in the basket. There are 22 red and green peaches in the basket. How many red peaches are in the basket?<|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
There are some red peaches, 90 yellow peaches, and 16 green peaches in the basket. There are 22 red and green peaches in the basket. How many red peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_57><Z_65><Z_86><Z_86><Z_31><Z_98><Z_73><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some red peaches, 90 yellow peaches, and 16 green peaches in the basket. There are 22 red and green peaches in the basket. How many red peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some red peaches, 90 yellow peaches, and 16 green peaches in the basket. There are 22 red and green peaches in the basket. How many red peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_60><Z_26><Z_67><Z_70><Z_81><Z_55><Z_1><Z_72><ANSWER>

Question: There are some red peaches, 90 yellow peaches, and 16 green peaches in the basket. There are 22 red and green peaches in the basket. How many red peaches are in the basket?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 17 persons in an office is 15 years. Out of these, the average age of 5 of them is 14 years and that of the other 9 persons is 16 years. What is the age of the 15th 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 average age of 17 persons in an office is 15 years. Out of these, the average age of 5 of them is 14 years and that of the other 9 persons is 16 years. What is the age of the 15th person?<|im_end|>
<|im_start|>assistant
<Z_94><Z_47><Z_4><Z_72><Z_70><Z_33><Z_66><Z_26><Z_78><Z_98><Z_66><Z_33><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 17 persons in an office is 15 years. Out of these, the average age of 5 of them is 14 years and that of the other 9 persons is 16 years. What is the age of the 15th person?<|im_end|>
<|im_start|>assistant
<Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 17 persons in an office is 15 years. Out of these, the average age of 5 of them is 14 years and that of the other 9 persons is 16 years. What is the age of the 15th person?<|im_end|>
<|im_start|>assistant
<Z_49><Z_91><Z_4><Z_90><Z_79><Z_95><Z_51><Z_75><Z_80><Z_112><Z_98><Z_29><Z_5><ANSWER>

Question: The average age of 17 persons in an office is 15 years. Out of these, the average age of 5 of them is 14 years and that of the other 9 persons is 16 years. What is the age of the 15th person?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gunther needs to clean his apartment. It takes him 45 minutes to vacuum the carpets, 60 minutes to dust the furniture, 30 minutes to mop the floors in his kitchen, and 5 minutes to brush each cat, and he has three cats. He has 3 hours of free time available, and he uses this time to clean his apartment. How many minutes of free time will he have left after he cleans the apartment?<|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
Gunther needs to clean his apartment. It takes him 45 minutes to vacuum the carpets, 60 minutes to dust the furniture, 30 minutes to mop the floors in his kitchen, and 5 minutes to brush each cat, and he has three cats. He has 3 hours of free time available, and he uses this time to clean his apartment. How many minutes of free time will he have left after he cleans the apartment?<|im_end|>
<|im_start|>assistant
<Z_94><Z_89><Z_107><Z_12><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gunther needs to clean his apartment. It takes him 45 minutes to vacuum the carpets, 60 minutes to dust the furniture, 30 minutes to mop the floors in his kitchen, and 5 minutes to brush each cat, and he has three cats. He has 3 hours of free time available, and he uses this time to clean his apartment. How many minutes of free time will he have left after he cleans the apartment?<|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
Gunther needs to clean his apartment. It takes him 45 minutes to vacuum the carpets, 60 minutes to dust the furniture, 30 minutes to mop the floors in his kitchen, and 5 minutes to brush each cat, and he has three cats. He has 3 hours of free time available, and he uses this time to clean his apartment. How many minutes of free time will he have left after he cleans the apartment?<|im_end|>
<|im_start|>assistant
<Z_24><Z_38><Z_117><Z_97><Z_9><ANSWER>

Question: Gunther needs to clean his apartment. It takes him 45 minutes to vacuum the carpets, 60 minutes to dust the furniture, 30 minutes to mop the floors in his kitchen, and 5 minutes to brush each cat, and he has three cats. He has 3 hours of free time available, and he uses this time to clean his apartment. How many minutes of free time will he have left after he cleans the apartment?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the remainder when $5^{5^{5^5}}$ is divided by 500.<|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 remainder when $5^{5^{5^5}}$ is divided by 500.<|im_end|>
<|im_start|>assistant
<Z_57><Z_103><Z_68><Z_126><Z_65><Z_126><Z_12><Z_49><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the remainder when $5^{5^{5^5}}$ is divided by 500.<|im_end|>
<|im_start|>assistant
<Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the remainder when $5^{5^{5^5}}$ is divided by 500.<|im_end|>
<|im_start|>assistant
<Z_84><Z_93><Z_85><Z_89><Z_2><Z_76><Z_108><Z_93><Z_7><ANSWER>

Question: Find the remainder when $5^{5^{5^5}}$ is divided by 500.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive integers, not exceeding 150, are multiples of 3 or 4 but not 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
How many positive integers, not exceeding 150, are multiples of 3 or 4 but not 6?<|im_end|>
<|im_start|>assistant
<Z_104><Z_37><Z_49><Z_100><Z_65><Z_98><Z_49><Z_33><Z_32><Z_28><Z_49><Z_91><Z_83><Z_126><Z_72><Z_53><Z_26><Z_126><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive integers, not exceeding 150, are multiples of 3 or 4 but not 6?<|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
How many positive integers, not exceeding 150, are multiples of 3 or 4 but not 6?<|im_end|>
<|im_start|>assistant
<Z_84><Z_25><Z_13><Z_120><Z_36><Z_75><Z_78><Z_81><Z_11><Z_127><Z_37><Z_28><Z_27><Z_57><Z_78><Z_88><Z_69><Z_73><Z_58><ANSWER>

Question: How many positive integers, not exceeding 150, are multiples of 3 or 4 but not 6?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If you cut and used 105 millimeters (mm) from a 23.3 centimeter (cm) wire and another 4.6 centimeters (cm) the next day, how many millimeters (mm) of wire are left?<|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 you cut and used 105 millimeters (mm) from a 23.3 centimeter (cm) wire and another 4.6 centimeters (cm) the next day, how many millimeters (mm) of wire are left?<|im_end|>
<|im_start|>assistant
<Z_94><Z_34><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If you cut and used 105 millimeters (mm) from a 23.3 centimeter (cm) wire and another 4.6 centimeters (cm) the next day, how many millimeters (mm) of wire are left?<|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
If you cut and used 105 millimeters (mm) from a 23.3 centimeter (cm) wire and another 4.6 centimeters (cm) the next day, how many millimeters (mm) of wire are left?<|im_end|>
<|im_start|>assistant
<Z_94><Z_78><Z_26><ANSWER>

Question: If you cut and used 105 millimeters (mm) from a 23.3 centimeter (cm) wire and another 4.6 centimeters (cm) the next day, how many millimeters (mm) of wire are left?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m = 3^some exponent, what is the greatest value of the exponent for which m is a factor of 25!?<|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
If m = 3^some exponent, what is the greatest value of the exponent for which m is a factor of 25!?<|im_end|>
<|im_start|>assistant
<Z_113><Z_107><Z_72><Z_93><Z_32><Z_56><Z_87><Z_12><Z_53><Z_70><Z_65><Z_8><Z_81><Z_92><Z_53><Z_52><Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m = 3^some exponent, what is the greatest value of the exponent for which m is a factor of 25!?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m = 3^some exponent, what is the greatest value of the exponent for which m is a factor of 25!?<|im_end|>
<|im_start|>assistant
<Z_73><Z_95><Z_68><Z_103><Z_20><Z_76><Z_66><Z_19><Z_85><Z_84><Z_111><Z_109><Z_82><Z_98><Z_79><Z_69><Z_118><ANSWER>

Question: If m = 3^some exponent, what is the greatest value of the exponent for which m is a factor of 25!?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a factory produces 4560 toys per week . if the workers at this factory work 4 days a week and if these workers make the same number of toys everyday , how many toys are produced each day ?<|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 factory produces 4560 toys per week . if the workers at this factory work 4 days a week and if these workers make the same number of toys everyday , how many toys are produced each day ?<|im_end|>
<|im_start|>assistant
<Z_29><Z_78><Z_124><Z_40><Z_90><Z_125><Z_34><Z_126><Z_119><Z_60><Z_96><Z_123><Z_70><Z_55><Z_83><Z_90><Z_63><Z_55><Z_37><Z_26><Z_60><Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a factory produces 4560 toys per week . if the workers at this factory work 4 days a week and if these workers make the same number of toys everyday , how many toys are produced each day ?<|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 factory produces 4560 toys per week . if the workers at this factory work 4 days a week and if these workers make the same number of toys everyday , how many toys are produced each day ?<|im_end|>
<|im_start|>assistant
<Z_109><Z_105><Z_102><Z_73><Z_65><Z_69><Z_39><Z_41><Z_83><Z_49><Z_11><Z_99><Z_62><Z_59><Z_99><Z_1><Z_44><Z_34><Z_30><Z_68><Z_3><Z_70><ANSWER>

Question: a factory produces 4560 toys per week . if the workers at this factory work 4 days a week and if these workers make the same number of toys everyday , how many toys are produced each day ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the 25% of a certain amount which is Rs. 200. What is the amount?<|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 25% of a certain amount which is Rs. 200. What is the amount?<|im_end|>
<|im_start|>assistant
<Z_104><Z_120><Z_77><Z_78><Z_22><Z_64><Z_39><Z_78><Z_114><Z_117><Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the 25% of a certain amount which is Rs. 200. What is the amount?<|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
Find the 25% of a certain amount which is Rs. 200. What is the amount?<|im_end|>
<|im_start|>assistant
<Z_79><Z_76><Z_47><Z_19><Z_53><Z_31><Z_23><Z_33><Z_43><Z_12><Z_109><ANSWER>

Question: Find the 25% of a certain amount which is Rs. 200. What is the amount?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the smallest possible product of two distinct prime numbers, each greater than 50.<|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
Determine the smallest possible product of two distinct prime numbers, each greater than 50.<|im_end|>
<|im_start|>assistant
<Z_94><Z_40><Z_37><Z_98><Z_59><Z_115><Z_124><Z_98><Z_37><Z_93><Z_31><Z_31><Z_76><Z_37><Z_120><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the smallest possible product of two distinct prime numbers, each greater than 50.<|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
Determine the smallest possible product of two distinct prime numbers, each greater than 50.<|im_end|>
<|im_start|>assistant
<Z_50><Z_65><Z_98><Z_62><Z_26><Z_6><Z_41><Z_92><Z_58><Z_15><Z_36><Z_8><Z_43><Z_46><Z_36><Z_101><ANSWER>

Question: Determine the smallest possible product of two distinct prime numbers, each greater than 50.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family buys 5 black pencils, 3 blue pens, 1 red pen, and 1 green pen. How many ways are there to arrange these writing utensils in a circular holder if the 2 blue pens are not to be adjacent?<|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
A family buys 5 black pencils, 3 blue pens, 1 red pen, and 1 green pen. How many ways are there to arrange these writing utensils in a circular holder if the 2 blue pens are not to be adjacent?<|im_end|>
<|im_start|>assistant
<Z_104><Z_33><Z_64><Z_12><Z_121><Z_59><Z_78><Z_80><Z_107><Z_37><Z_48><Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family buys 5 black pencils, 3 blue pens, 1 red pen, and 1 green pen. How many ways are there to arrange these writing utensils in a circular holder if the 2 blue pens are not to be adjacent?<|im_end|>
<|im_start|>assistant
<Z_35><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family buys 5 black pencils, 3 blue pens, 1 red pen, and 1 green pen. How many ways are there to arrange these writing utensils in a circular holder if the 2 blue pens are not to be adjacent?<|im_end|>
<|im_start|>assistant
<Z_25><Z_61><Z_19><Z_124><Z_96><Z_66><Z_76><Z_78><Z_65><Z_94><Z_39><Z_45><ANSWER>

Question: A family buys 5 black pencils, 3 blue pens, 1 red pen, and 1 green pen. How many ways are there to arrange these writing utensils in a circular holder if the 2 blue pens are not to be adjacent?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 34 pencils and 49 crayons in the drawer. Dan took 22 pencils from the drawer and then returned 5 of them. After that, he took 11 crayons out of the drawer. How many pencils are now in the drawer?<|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
There are 34 pencils and 49 crayons in the drawer. Dan took 22 pencils from the drawer and then returned 5 of them. After that, he took 11 crayons out of the drawer. How many pencils are now in the drawer?<|im_end|>
<|im_start|>assistant
<Z_57><Z_1><Z_63><Z_119><Z_31><Z_47><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 34 pencils and 49 crayons in the drawer. Dan took 22 pencils from the drawer and then returned 5 of them. After that, he took 11 crayons out of the drawer. How many pencils are now in the drawer?<|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
There are 34 pencils and 49 crayons in the drawer. Dan took 22 pencils from the drawer and then returned 5 of them. After that, he took 11 crayons out of the drawer. How many pencils are now in the drawer?<|im_end|>
<|im_start|>assistant
<Z_18><Z_64><Z_36><Z_45><Z_22><Z_89><Z_25><ANSWER>

Question: There are 34 pencils and 49 crayons in the drawer. Dan took 22 pencils from the drawer and then returned 5 of them. After that, he took 11 crayons out of the drawer. How many pencils are now in the drawer?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a castle, there are 25 identical square rooms forming a $5 \times 5$ square. Each of the 25 rooms is occupied by one person - either a liar or a knight (liars always lie, knights always tell the truth). Each of these 25 people said, "In at least one of the rooms adjacent to mine, there lives a liar." Rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 25 people?<|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 castle, there are 25 identical square rooms forming a $5 \times 5$ square. Each of the 25 rooms is occupied by one person - either a liar or a knight (liars always lie, knights always tell the truth). Each of these 25 people said, "In at least one of the rooms adjacent to mine, there lives a liar." Rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 25 people?<|im_end|>
<|im_start|>assistant
<Z_104><Z_86><Z_44><Z_121><Z_68><Z_8><Z_93><Z_56><Z_78><Z_117><Z_26><Z_10><Z_63><Z_48><Z_87><Z_88><Z_12><Z_60><Z_123><Z_92><Z_55><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a castle, there are 25 identical square rooms forming a $5 \times 5$ square. Each of the 25 rooms is occupied by one person - either a liar or a knight (liars always lie, knights always tell the truth). Each of these 25 people said, "In at least one of the rooms adjacent to mine, there lives a liar." Rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 25 people?<|im_end|>
<|im_start|>assistant
<Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a castle, there are 25 identical square rooms forming a $5 \times 5$ square. Each of the 25 rooms is occupied by one person - either a liar or a knight (liars always lie, knights always tell the truth). Each of these 25 people said, "In at least one of the rooms adjacent to mine, there lives a liar." Rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 25 people?<|im_end|>
<|im_start|>assistant
<Z_104><Z_53><Z_36><Z_118><Z_40><Z_6><Z_35><Z_97><Z_55><Z_59><Z_45><Z_38><Z_111><Z_16><Z_88><Z_68><Z_17><Z_57><Z_53><Z_65><Z_26><ANSWER>

Question: In a castle, there are 25 identical square rooms forming a $5 \times 5$ square. Each of the 25 rooms is occupied by one person - either a liar or a knight (liars always lie, knights always tell the truth). Each of these 25 people said, "In at least one of the rooms adjacent to mine, there lives a liar." Rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 25 people?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?<|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 light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?<|im_end|>
<|im_start|>assistant
<Z_104><Z_91><Z_97><Z_87><Z_52><Z_93><Z_90><Z_59><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?<|im_end|>
<|im_start|>assistant
<Z_42><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?<|im_end|>
<|im_start|>assistant
<Z_29><Z_3><Z_125><Z_24><Z_13><Z_85><Z_110><Z_109><Z_101><ANSWER>

Question: The light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the square root of a positive number $x$ is $3a-4$ and $1-6a$, then $a=$______, $x=______.<|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 square root of a positive number $x$ is $3a-4$ and $1-6a$, then $a=$______, $x=______.<|im_end|>
<|im_start|>assistant
<Z_104><Z_68><Z_34><Z_124><Z_81><Z_82><Z_103><Z_96><Z_61><Z_101><Z_119><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the square root of a positive number $x$ is $3a-4$ and $1-6a$, then $a=$______, $x=______.<|im_end|>
<|im_start|>assistant
<Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the square root of a positive number $x$ is $3a-4$ and $1-6a$, then $a=$______, $x=______.<|im_end|>
<|im_start|>assistant
<Z_80><Z_72><Z_81><Z_22><Z_64><Z_0><Z_35><Z_25><Z_33><Z_33><Z_12><Z_37><ANSWER>

Question: If the square root of a positive number $x$ is $3a-4$ and $1-6a$, then $a=$______, $x=______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of intersection points between the graphs of $r = 3 \cos \theta$ and $r = 6 \sin \theta$.<|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
Determine the number of intersection points between the graphs of $r = 3 \cos \theta$ and $r = 6 \sin \theta$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_72><Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of intersection points between the graphs of $r = 3 \cos \theta$ and $r = 6 \sin \theta$.<|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
Determine the number of intersection points between the graphs of $r = 3 \cos \theta$ and $r = 6 \sin \theta$.<|im_end|>
<|im_start|>assistant
<Z_105><Z_113><Z_22><ANSWER>

Question: Determine the number of intersection points between the graphs of $r = 3 \cos \theta$ and $r = 6 \sin \theta$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Is there a positive integer $m$, such that for any positive integer $n$, $f(n)={3}^{n}(2n+7)+m$ is divisible by $36$? If it exists, find the minimum value of $m$ and prove your conclusion using mathematical induction; if it does not exist, please explain why.<|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
Is there a positive integer $m$, such that for any positive integer $n$, $f(n)={3}^{n}(2n+7)+m$ is divisible by $36$? If it exists, find the minimum value of $m$ and prove your conclusion using mathematical induction; if it does not exist, please explain why.<|im_end|>
<|im_start|>assistant
<Z_57><Z_89><Z_59><Z_86><Z_58><Z_97><Z_114><Z_82><Z_97><Z_53><Z_11><Z_91><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Is there a positive integer $m$, such that for any positive integer $n$, $f(n)={3}^{n}(2n+7)+m$ is divisible by $36$? If it exists, find the minimum value of $m$ and prove your conclusion using mathematical induction; if it does not exist, please explain why.<|im_end|>
<|im_start|>assistant
<Z_23><Z_96><Z_64><Z_13><Z_29><Z_11><Z_11><Z_70><Z_46><Z_18><Z_98><Z_22><Z_44><ANSWER>

Question: Is there a positive integer $m$, such that for any positive integer $n$, $f(n)={3}^{n}(2n+7)+m$ is divisible by $36$? If it exists, find the minimum value of $m$ and prove your conclusion using mathematical induction; if it does not exist, please explain why.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school arranges Chinese, Mathematics, English, Physics, Chemistry, elective courses, and 2 self-study periods within 8 class periods in one day. The first period can only be one of Chinese, Mathematics, or English. The eighth period can only be an elective course or a self-study period. Elective courses and self-study periods, as well as self-study periods with each other, cannot be adjacent. The total number of different arrangements is \_\_\_\_\_\_ (The result should be expressed in numbers).<|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
A school arranges Chinese, Mathematics, English, Physics, Chemistry, elective courses, and 2 self-study periods within 8 class periods in one day. The first period can only be one of Chinese, Mathematics, or English. The eighth period can only be an elective course or a self-study period. Elective courses and self-study periods, as well as self-study periods with each other, cannot be adjacent. The total number of different arrangements is \_\_\_\_\_\_ (The result should be expressed in numbers).<|im_end|>
<|im_start|>assistant
<Z_57><Z_86><Z_119><Z_93><Z_121><Z_16><Z_30><Z_78><Z_22><Z_11><Z_99><Z_49><Z_32><Z_71><Z_124><Z_90><Z_22><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school arranges Chinese, Mathematics, English, Physics, Chemistry, elective courses, and 2 self-study periods within 8 class periods in one day. The first period can only be one of Chinese, Mathematics, or English. The eighth period can only be an elective course or a self-study period. Elective courses and self-study periods, as well as self-study periods with each other, cannot be adjacent. The total number of different arrangements is \_\_\_\_\_\_ (The result should be expressed in numbers).<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school arranges Chinese, Mathematics, English, Physics, Chemistry, elective courses, and 2 self-study periods within 8 class periods in one day. The first period can only be one of Chinese, Mathematics, or English. The eighth period can only be an elective course or a self-study period. Elective courses and self-study periods, as well as self-study periods with each other, cannot be adjacent. The total number of different arrangements is \_\_\_\_\_\_ (The result should be expressed in numbers).<|im_end|>
<|im_start|>assistant
<Z_12><Z_72><Z_55><Z_123><Z_49><Z_48><Z_64><Z_8><Z_56><Z_80><Z_72><Z_9><Z_10><Z_56><Z_56><Z_33><Z_95><Z_94><ANSWER>

Question: A school arranges Chinese, Mathematics, English, Physics, Chemistry, elective courses, and 2 self-study periods within 8 class periods in one day. The first period can only be one of Chinese, Mathematics, or English. The eighth period can only be an elective course or a self-study period. Elective courses and self-study periods, as well as self-study periods with each other, cannot be adjacent. The total number of different arrangements is \_\_\_\_\_\_ (The result should be expressed in numbers).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(This question is worth 7 points) A store is having a seasonal sale on its merchandise. If the merchandise is sold at 60% of its original price, the store will lose 20 yuan. If it is sold at 80% of its original price, the store will gain 15 yuan. The question is: What is the original price of this merchandise?<|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
(This question is worth 7 points) A store is having a seasonal sale on its merchandise. If the merchandise is sold at 60% of its original price, the store will lose 20 yuan. If it is sold at 80% of its original price, the store will gain 15 yuan. The question is: What is the original price of this merchandise?<|im_end|>
<|im_start|>assistant
<Z_113><Z_125><Z_44><Z_12><Z_119><Z_36><Z_126><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(This question is worth 7 points) A store is having a seasonal sale on its merchandise. If the merchandise is sold at 60% of its original price, the store will lose 20 yuan. If it is sold at 80% of its original price, the store will gain 15 yuan. The question is: What is the original price of this merchandise?<|im_end|>
<|im_start|>assistant
<Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(This question is worth 7 points) A store is having a seasonal sale on its merchandise. If the merchandise is sold at 60% of its original price, the store will lose 20 yuan. If it is sold at 80% of its original price, the store will gain 15 yuan. The question is: What is the original price of this merchandise?<|im_end|>
<|im_start|>assistant
<Z_8><Z_51><Z_105><Z_93><Z_104><Z_20><Z_72><Z_33><ANSWER>

Question: (This question is worth 7 points) A store is having a seasonal sale on its merchandise. If the merchandise is sold at 60% of its original price, the store will lose 20 yuan. If it is sold at 80% of its original price, the store will gain 15 yuan. The question is: What is the original price of this merchandise?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a\ast b = 4a + 5b - 2ab$, what is the value of $6\ast3$?<|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 $a\ast b = 4a + 5b - 2ab$, what is the value of $6\ast3$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_65><Z_62><Z_81><Z_99><Z_92><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a\ast b = 4a + 5b - 2ab$, what is the value of $6\ast3$?<|im_end|>
<|im_start|>assistant
<Z_17><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a\ast b = 4a + 5b - 2ab$, what is the value of $6\ast3$?<|im_end|>
<|im_start|>assistant
<Z_22><Z_97><Z_58><Z_27><Z_47><Z_41><Z_54><ANSWER>

Question: If $a\ast b = 4a + 5b - 2ab$, what is the value of $6\ast3$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One day Max says to Liz, "Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French class. How many people are taking English but not French?<|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
One day Max says to Liz, "Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French class. How many people are taking English but not French?<|im_end|>
<|im_start|>assistant
<Z_104><Z_30><Z_58><Z_59><Z_83><Z_31><Z_37><Z_11><Z_117><Z_28><Z_39><Z_12><Z_118><Z_39><Z_75><Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One day Max says to Liz, "Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French class. How many people are taking English but not French?<|im_end|>
<|im_start|>assistant
<Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One day Max says to Liz, "Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French class. How many people are taking English but not French?<|im_end|>
<|im_start|>assistant
<Z_46><Z_52><Z_12><Z_81><Z_120><Z_96><Z_59><Z_104><Z_58><Z_70><Z_92><Z_115><Z_90><Z_67><Z_2><Z_81><ANSWER>

Question: One day Max says to Liz, "Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French class. How many people are taking English but not French?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a triangle, there are generally nine altitudes, angle bisectors, and medians. Investigate cases when there are fewer.<|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 triangle, there are generally nine altitudes, angle bisectors, and medians. Investigate cases when there are fewer.<|im_end|>
<|im_start|>assistant
<Z_94><Z_115><Z_89><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a triangle, there are generally nine altitudes, angle bisectors, and medians. Investigate cases when there are fewer.<|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
In a triangle, there are generally nine altitudes, angle bisectors, and medians. Investigate cases when there are fewer.<|im_end|>
<|im_start|>assistant
<Z_37><Z_104><Z_64><Z_47><ANSWER>

Question: In a triangle, there are generally nine altitudes, angle bisectors, and medians. Investigate cases when there are fewer.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man purchased 4 cotton blankets @ Rs. 100 each, 3 woolen blankets @ Rs. 150 each, and 2 silk blankets at a certain rate which is now slipped off from his memory. In addition, there was a discount on the cotton and woolen blankets: the cotton blankets had a 10% discount, and the woolen blankets had a 5% discount. He also remembers that the average price of all the blankets, after applying the discounts but before adding any taxes, was Rs. 130. Find the original price of the two silk blankets before any discounts or taxes were applied.<|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
A man purchased 4 cotton blankets @ Rs. 100 each, 3 woolen blankets @ Rs. 150 each, and 2 silk blankets at a certain rate which is now slipped off from his memory. In addition, there was a discount on the cotton and woolen blankets: the cotton blankets had a 10% discount, and the woolen blankets had a 5% discount. He also remembers that the average price of all the blankets, after applying the discounts but before adding any taxes, was Rs. 130. Find the original price of the two silk blankets before any discounts or taxes were applied.<|im_end|>
<|im_start|>assistant
<Z_94><Z_52><Z_27><Z_49><Z_73><Z_31><Z_16><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man purchased 4 cotton blankets @ Rs. 100 each, 3 woolen blankets @ Rs. 150 each, and 2 silk blankets at a certain rate which is now slipped off from his memory. In addition, there was a discount on the cotton and woolen blankets: the cotton blankets had a 10% discount, and the woolen blankets had a 5% discount. He also remembers that the average price of all the blankets, after applying the discounts but before adding any taxes, was Rs. 130. Find the original price of the two silk blankets before any discounts or taxes were applied.<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man purchased 4 cotton blankets @ Rs. 100 each, 3 woolen blankets @ Rs. 150 each, and 2 silk blankets at a certain rate which is now slipped off from his memory. In addition, there was a discount on the cotton and woolen blankets: the cotton blankets had a 10% discount, and the woolen blankets had a 5% discount. He also remembers that the average price of all the blankets, after applying the discounts but before adding any taxes, was Rs. 130. Find the original price of the two silk blankets before any discounts or taxes were applied.<|im_end|>
<|im_start|>assistant
<Z_93><Z_116><Z_54><Z_76><Z_47><Z_111><Z_33><Z_18><ANSWER>

Question: A man purchased 4 cotton blankets @ Rs. 100 each, 3 woolen blankets @ Rs. 150 each, and 2 silk blankets at a certain rate which is now slipped off from his memory. In addition, there was a discount on the cotton and woolen blankets: the cotton blankets had a 10% discount, and the woolen blankets had a 5% discount. He also remembers that the average price of all the blankets, after applying the discounts but before adding any taxes, was Rs. 130. Find the original price of the two silk blankets before any discounts or taxes were applied.