<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fly moves from the origin only to the right or upwards along the lines of an integer grid (monotonic movement). At each grid node, the fly randomly chooses the direction of further movement: upwards or to the right.

a) Prove that, sooner or later, the fly will reach the point with the x-coordinate 2011.

b) Find the expected value of the fly's y-coordinate at the moment when the fly reaches the x-coordinate 2011.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fly moves from the origin only to the right or upwards along the lines of an integer grid (monotonic movement). At each grid node, the fly randomly chooses the direction of further movement: upwards or to the right.

a) Prove that, sooner or later, the fly will reach the point with the x-coordinate 2011.

b) Find the expected value of the fly's y-coordinate at the moment when the fly reaches the x-coordinate 2011.<|im_end|>
<|im_start|>assistant
<Z_24><Z_509><Z_462><Z_172><Z_97><Z_246><Z_200><Z_87><Z_200><Z_328><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fly moves from the origin only to the right or upwards along the lines of an integer grid (monotonic movement). At each grid node, the fly randomly chooses the direction of further movement: upwards or to the right.

a) Prove that, sooner or later, the fly will reach the point with the x-coordinate 2011.

b) Find the expected value of the fly's y-coordinate at the moment when the fly reaches the x-coordinate 2011.<|im_end|>
<|im_start|>assistant
<Z_378><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fly moves from the origin only to the right or upwards along the lines of an integer grid (monotonic movement). At each grid node, the fly randomly chooses the direction of further movement: upwards or to the right.

a) Prove that, sooner or later, the fly will reach the point with the x-coordinate 2011.

b) Find the expected value of the fly's y-coordinate at the moment when the fly reaches the x-coordinate 2011.<|im_end|>
<|im_start|>assistant
<Z_271><Z_164><Z_56><Z_413><Z_46><Z_123><Z_102><Z_136><Z_329><Z_169><Z_195><ANSWER>

Question:
A fly moves from the origin only to the right or upwards along the lines of an integer grid (monotonic movement). At each grid node, the fly randomly chooses the direction of further movement: upwards or to the right.

a) Prove that, sooner or later, the fly will reach the point with the x-coordinate 2011.

b) Find the expected value of the fly's y-coordinate at the moment when the fly reaches the x-coordinate 2011.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From unit cubes, a large parallelepiped with side lengths greater than 3 was assembled. Two cubes are called neighboring if they share a face. Therefore, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From unit cubes, a large parallelepiped with side lengths greater than 3 was assembled. Two cubes are called neighboring if they share a face. Therefore, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.<|im_end|>
<|im_start|>assistant
<Z_477><Z_310><Z_186><Z_344><Z_319><Z_467><Z_290><Z_247><Z_443><Z_97><Z_418><Z_462><Z_94><Z_441><Z_423><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From unit cubes, a large parallelepiped with side lengths greater than 3 was assembled. Two cubes are called neighboring if they share a face. Therefore, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.<|im_end|>
<|im_start|>assistant
<Z_333><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From unit cubes, a large parallelepiped with side lengths greater than 3 was assembled. Two cubes are called neighboring if they share a face. Therefore, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.<|im_end|>
<|im_start|>assistant
<Z_282><Z_133><Z_178><Z_298><Z_15><Z_353><Z_434><Z_451><Z_112><Z_452><Z_413><Z_390><Z_381><Z_31><Z_34><ANSWER>

Question: From unit cubes, a large parallelepiped with side lengths greater than 3 was assembled. Two cubes are called neighboring if they share a face. Therefore, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The prime factorization of 2014 is $2 \times 19 \times 53$. How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy = 2014$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The prime factorization of 2014 is $2 \times 19 \times 53$. How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy = 2014$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_232><Z_81><Z_191><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The prime factorization of 2014 is $2 \times 19 \times 53$. How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy = 2014$?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The prime factorization of 2014 is $2 \times 19 \times 53$. How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy = 2014$?<|im_end|>
<|im_start|>assistant
<Z_66><Z_255><Z_34><Z_96><Z_362><ANSWER>

Question: The prime factorization of 2014 is $2 \times 19 \times 53$. How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy = 2014$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at a certain speed crosses a pole in 15 seconds. The length of the train is 375 meters. What is the speed of the train in km/hr?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at a certain speed crosses a pole in 15 seconds. The length of the train is 375 meters. What is the speed of the train in km/hr?<|im_end|>
<|im_start|>assistant
<Z_24><Z_34><Z_250><Z_295><Z_150><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at a certain speed crosses a pole in 15 seconds. The length of the train is 375 meters. What is the speed of the train in km/hr?<|im_end|>
<|im_start|>assistant
<Z_116><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train running at a certain speed crosses a pole in 15 seconds. The length of the train is 375 meters. What is the speed of the train in km/hr?<|im_end|>
<|im_start|>assistant
<Z_28><Z_268><Z_63><Z_209><Z_266><ANSWER>

Question: A train running at a certain speed crosses a pole in 15 seconds. The length of the train is 375 meters. What is the speed of the train in km/hr?


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

The volume of a tetrahedron and a pyramid

A convex polyhedron \(ABCDEF\) has five faces: \(CDF\), \(ABE\), \(BCFE\), \(ADF E\), and \(ABCD\). Edge \(AB\) is parallel to edge \(CD\). Points \(K\) and \(L\) are located on edges \(AD\) and \(BC\), respectively, such that segment \(KL\) divides the area of face \(ABCD\) in half. Point \(M\) is the midpoint of edge \(EF\) and is the apex of the pyramid \(MABCD\), whose volume is 6. Find the volume of pyramid \(EKLF\) if the volume of polyhedron \(ABCDEF\) is 19.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

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

The volume of a tetrahedron and a pyramid

A convex polyhedron \(ABCDEF\) has five faces: \(CDF\), \(ABE\), \(BCFE\), \(ADF E\), and \(ABCD\). Edge \(AB\) is parallel to edge \(CD\). Points \(K\) and \(L\) are located on edges \(AD\) and \(BC\), respectively, such that segment \(KL\) divides the area of face \(ABCD\) in half. Point \(M\) is the midpoint of edge \(EF\) and is the apex of the pyramid \(MABCD\), whose volume is 6. Find the volume of pyramid \(EKLF\) if the volume of polyhedron \(ABCDEF\) is 19.<|im_end|>
<|im_start|>assistant
<Z_369><Z_211><Z_238><Z_420><Z_219><Z_363><ANSWER>

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

The volume of a tetrahedron and a pyramid

A convex polyhedron \(ABCDEF\) has five faces: \(CDF\), \(ABE\), \(BCFE\), \(ADF E\), and \(ABCD\). Edge \(AB\) is parallel to edge \(CD\). Points \(K\) and \(L\) are located on edges \(AD\) and \(BC\), respectively, such that segment \(KL\) divides the area of face \(ABCD\) in half. Point \(M\) is the midpoint of edge \(EF\) and is the apex of the pyramid \(MABCD\), whose volume is 6. Find the volume of pyramid \(EKLF\) if the volume of polyhedron \(ABCDEF\) is 19.<|im_end|>
<|im_start|>assistant
<Z_179><ANSWER>

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

The volume of a tetrahedron and a pyramid

A convex polyhedron \(ABCDEF\) has five faces: \(CDF\), \(ABE\), \(BCFE\), \(ADF E\), and \(ABCD\). Edge \(AB\) is parallel to edge \(CD\). Points \(K\) and \(L\) are located on edges \(AD\) and \(BC\), respectively, such that segment \(KL\) divides the area of face \(ABCD\) in half. Point \(M\) is the midpoint of edge \(EF\) and is the apex of the pyramid \(MABCD\), whose volume is 6. Find the volume of pyramid \(EKLF\) if the volume of polyhedron \(ABCDEF\) is 19.<|im_end|>
<|im_start|>assistant
<Z_216><Z_405><Z_320><Z_285><Z_271><Z_414><ANSWER>

Question:
The volume of a tetrahedron and a pyramid

A convex polyhedron \(ABCDEF\) has five faces: \(CDF\), \(ABE\), \(BCFE\), \(ADF E\), and \(ABCD\). Edge \(AB\) is parallel to edge \(CD\). Points \(K\) and \(L\) are located on edges \(AD\) and \(BC\), respectively, such that segment \(KL\) divides the area of face \(ABCD\) in half. Point \(M\) is the midpoint of edge \(EF\) and is the apex of the pyramid \(MABCD\), whose volume is 6. Find the volume of pyramid \(EKLF\) if the volume of polyhedron \(ABCDEF\) is 19.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The grasshopper, the frog, the mouse, and the squirrel had a jumping contest with obstacles. The grasshopper jumped 19 inches, but it encountered an obstacle which reduced its jump distance by 3 inches. The frog jumped 10 inches farther than the grasshopper without encountering any obstacles. The mouse jumped 20 inches farther than the frog, but encountered an obstacle that reduced its jump distance by 5 inches. The squirrel jumped 7 inches less than the mouse, and it faced a 2-inch obstacle. How much farther did the mouse jump than the grasshopper, considering the obstacles they encountered?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The grasshopper, the frog, the mouse, and the squirrel had a jumping contest with obstacles. The grasshopper jumped 19 inches, but it encountered an obstacle which reduced its jump distance by 3 inches. The frog jumped 10 inches farther than the grasshopper without encountering any obstacles. The mouse jumped 20 inches farther than the frog, but encountered an obstacle that reduced its jump distance by 5 inches. The squirrel jumped 7 inches less than the mouse, and it faced a 2-inch obstacle. How much farther did the mouse jump than the grasshopper, considering the obstacles they encountered?<|im_end|>
<|im_start|>assistant
<Z_477><Z_63><Z_482><Z_97><Z_274><Z_292><Z_260><Z_248><Z_346><Z_77><Z_344><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The grasshopper, the frog, the mouse, and the squirrel had a jumping contest with obstacles. The grasshopper jumped 19 inches, but it encountered an obstacle which reduced its jump distance by 3 inches. The frog jumped 10 inches farther than the grasshopper without encountering any obstacles. The mouse jumped 20 inches farther than the frog, but encountered an obstacle that reduced its jump distance by 5 inches. The squirrel jumped 7 inches less than the mouse, and it faced a 2-inch obstacle. How much farther did the mouse jump than the grasshopper, considering the obstacles they encountered?<|im_end|>
<|im_start|>assistant
<Z_240><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The grasshopper, the frog, the mouse, and the squirrel had a jumping contest with obstacles. The grasshopper jumped 19 inches, but it encountered an obstacle which reduced its jump distance by 3 inches. The frog jumped 10 inches farther than the grasshopper without encountering any obstacles. The mouse jumped 20 inches farther than the frog, but encountered an obstacle that reduced its jump distance by 5 inches. The squirrel jumped 7 inches less than the mouse, and it faced a 2-inch obstacle. How much farther did the mouse jump than the grasshopper, considering the obstacles they encountered?<|im_end|>
<|im_start|>assistant
<Z_154><Z_430><Z_36><Z_412><Z_472><Z_37><Z_372><Z_444><Z_480><Z_90><Z_128><ANSWER>

Question: The grasshopper, the frog, the mouse, and the squirrel had a jumping contest with obstacles. The grasshopper jumped 19 inches, but it encountered an obstacle which reduced its jump distance by 3 inches. The frog jumped 10 inches farther than the grasshopper without encountering any obstacles. The mouse jumped 20 inches farther than the frog, but encountered an obstacle that reduced its jump distance by 5 inches. The squirrel jumped 7 inches less than the mouse, and it faced a 2-inch obstacle. How much farther did the mouse jump than the grasshopper, considering the obstacles they encountered?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the area of a rhombus whose two diagonals are 14 centimeters (cm) and 24 centimeters (cm).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the area of a rhombus whose two diagonals are 14 centimeters (cm) and 24 centimeters (cm).<|im_end|>
<|im_start|>assistant
<Z_369><Z_68><Z_326><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the area of a rhombus whose two diagonals are 14 centimeters (cm) and 24 centimeters (cm).<|im_end|>
<|im_start|>assistant
<Z_355><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the area of a rhombus whose two diagonals are 14 centimeters (cm) and 24 centimeters (cm).<|im_end|>
<|im_start|>assistant
<Z_410><Z_301><Z_109><ANSWER>

Question: Find the area of a rhombus whose two diagonals are 14 centimeters (cm) and 24 centimeters (cm).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the remainder when $4 \times 14 \times 24 \times 34 \times \ldots \times 184 \times 194$ is divided by $5$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the remainder when $4 \times 14 \times 24 \times 34 \times \ldots \times 184 \times 194$ is divided by $5$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_144><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 $4 \times 14 \times 24 \times 34 \times \ldots \times 184 \times 194$ is divided by $5$.<|im_end|>
<|im_start|>assistant
<Z_13><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 $4 \times 14 \times 24 \times 34 \times \ldots \times 184 \times 194$ is divided by $5$.<|im_end|>
<|im_start|>assistant
<Z_138><Z_253><ANSWER>

Question: Find the remainder when $4 \times 14 \times 24 \times 34 \times \ldots \times 184 \times 194$ is divided by $5$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A worksheet had some problems on it. A teacher had 16 worksheets to grade and had already graded 8 of them. She has 32 more problems to grade. How many problems are on each worksheet?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A worksheet had some problems on it. A teacher had 16 worksheets to grade and had already graded 8 of them. She has 32 more problems to grade. How many problems are on each worksheet?<|im_end|>
<|im_start|>assistant
<Z_477><Z_509><Z_304><Z_86><Z_343><Z_357><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A worksheet had some problems on it. A teacher had 16 worksheets to grade and had already graded 8 of them. She has 32 more problems to grade. How many problems are on each worksheet?<|im_end|>
<|im_start|>assistant
<Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A worksheet had some problems on it. A teacher had 16 worksheets to grade and had already graded 8 of them. She has 32 more problems to grade. How many problems are on each worksheet?<|im_end|>
<|im_start|>assistant
<Z_268><Z_14><Z_7><Z_392><Z_426><Z_426><ANSWER>

Question: A worksheet had some problems on it. A teacher had 16 worksheets to grade and had already graded 8 of them. She has 32 more problems to grade. How many problems are on each worksheet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a is two years older than b, who is twice as old as c, and d is three years younger than a. If the total of the ages of a, b, c, and d is 44, then how old is b?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a is two years older than b, who is twice as old as c, and d is three years younger than a. If the total of the ages of a, b, c, and d is 44, then how old is b?<|im_end|>
<|im_start|>assistant
<Z_369><Z_496><Z_410><Z_213><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a is two years older than b, who is twice as old as c, and d is three years younger than a. If the total of the ages of a, b, c, and d is 44, then how old is b?<|im_end|>
<|im_start|>assistant
<Z_448><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a is two years older than b, who is twice as old as c, and d is three years younger than a. If the total of the ages of a, b, c, and d is 44, then how old is b?<|im_end|>
<|im_start|>assistant
<Z_104><Z_151><Z_353><Z_348><ANSWER>

Question: a is two years older than b, who is twice as old as c, and d is three years younger than a. If the total of the ages of a, b, c, and d is 44, then how old is b?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $f(x)=\frac{ax^2+bx+c}{dx+b}$ with $abcd \neq 0$ and ensuring that no division by zero occurs within its domain, and $f(f(x)) = x$ for all $x$ for which $f$ is defined, what is the expression for $a+d$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $f(x)=\frac{ax^2+bx+c}{dx+b}$ with $abcd \neq 0$ and ensuring that no division by zero occurs within its domain, and $f(f(x)) = x$ for all $x$ for which $f$ is defined, what is the expression for $a+d$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_177><Z_355><Z_305><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $f(x)=\frac{ax^2+bx+c}{dx+b}$ with $abcd \neq 0$ and ensuring that no division by zero occurs within its domain, and $f(f(x)) = x$ for all $x$ for which $f$ is defined, what is the expression for $a+d$?<|im_end|>
<|im_start|>assistant
<Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $f(x)=\frac{ax^2+bx+c}{dx+b}$ with $abcd \neq 0$ and ensuring that no division by zero occurs within its domain, and $f(f(x)) = x$ for all $x$ for which $f$ is defined, what is the expression for $a+d$?<|im_end|>
<|im_start|>assistant
<Z_140><Z_406><Z_249><Z_371><ANSWER>

Question: If $f(x)=\frac{ax^2+bx+c}{dx+b}$ with $abcd \neq 0$ and ensuring that no division by zero occurs within its domain, and $f(f(x)) = x$ for all $x$ for which $f$ is defined, what is the expression for $a+d$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the number of terms in the sequences $\{a_{n}\}$ and $\{b_{n}\}$ be $m$. The distance between the sequences $\{a_{n}\}$ and $\{b_{n}\}$ is defined as $\sum_{i=1}^{m}|a_{i}-b_{i}|$.
$(Ⅰ)$ Find the distance between the sequences $1$, $4$, $6$, $7$ and $3$, $4$, $11$, $8$.
$(Ⅱ)$ Let $A$ be the set of all sequences $\{a_{n}\}$ that satisfy the recursive relation $a_{n+1}=\frac{1+a_{n}}{1-a_{n}}$. Let $\{b_{n}\}$ and $\{c_{n}\}$ be two elements in $A$, both with $m$ terms. If $b_{1}=2$, $c_{1}=3$, and the distance between $\{b_{n}\}$ and $\{c_{n}\}$ is less than $2016$, find the maximum value of $m$.
$(Ⅲ)$ Let $S$ be the set of all $7$-term sequences $\{a_{n}|1\leqslant n\leqslant 7$, $a_{n}=0$ or $1\}$. Let $T\subseteq S$ such that the distance between any two elements in $T$ is greater than or equal to $3$. Prove that the number of elements in $T$ is less than or equal to $16$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the number of terms in the sequences $\{a_{n}\}$ and $\{b_{n}\}$ be $m$. The distance between the sequences $\{a_{n}\}$ and $\{b_{n}\}$ is defined as $\sum_{i=1}^{m}|a_{i}-b_{i}|$.
$(Ⅰ)$ Find the distance between the sequences $1$, $4$, $6$, $7$ and $3$, $4$, $11$, $8$.
$(Ⅱ)$ Let $A$ be the set of all sequences $\{a_{n}\}$ that satisfy the recursive relation $a_{n+1}=\frac{1+a_{n}}{1-a_{n}}$. Let $\{b_{n}\}$ and $\{c_{n}\}$ be two elements in $A$, both with $m$ terms. If $b_{1}=2$, $c_{1}=3$, and the distance between $\{b_{n}\}$ and $\{c_{n}\}$ is less than $2016$, find the maximum value of $m$.
$(Ⅲ)$ Let $S$ be the set of all $7$-term sequences $\{a_{n}|1\leqslant n\leqslant 7$, $a_{n}=0$ or $1\}$. Let $T\subseteq S$ such that the distance between any two elements in $T$ is greater than or equal to $3$. Prove that the number of elements in $T$ is less than or equal to $16$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_264><Z_283><Z_253><Z_424><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let the number of terms in the sequences $\{a_{n}\}$ and $\{b_{n}\}$ be $m$. The distance between the sequences $\{a_{n}\}$ and $\{b_{n}\}$ is defined as $\sum_{i=1}^{m}|a_{i}-b_{i}|$.
$(Ⅰ)$ Find the distance between the sequences $1$, $4$, $6$, $7$ and $3$, $4$, $11$, $8$.
$(Ⅱ)$ Let $A$ be the set of all sequences $\{a_{n}\}$ that satisfy the recursive relation $a_{n+1}=\frac{1+a_{n}}{1-a_{n}}$. Let $\{b_{n}\}$ and $\{c_{n}\}$ be two elements in $A$, both with $m$ terms. If $b_{1}=2$, $c_{1}=3$, and the distance between $\{b_{n}\}$ and $\{c_{n}\}$ is less than $2016$, find the maximum value of $m$.
$(Ⅲ)$ Let $S$ be the set of all $7$-term sequences $\{a_{n}|1\leqslant n\leqslant 7$, $a_{n}=0$ or $1\}$. Let $T\subseteq S$ such that the distance between any two elements in $T$ is greater than or equal to $3$. Prove that the number of elements in $T$ is less than or equal to $16$.<|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
Let the number of terms in the sequences $\{a_{n}\}$ and $\{b_{n}\}$ be $m$. The distance between the sequences $\{a_{n}\}$ and $\{b_{n}\}$ is defined as $\sum_{i=1}^{m}|a_{i}-b_{i}|$.
$(Ⅰ)$ Find the distance between the sequences $1$, $4$, $6$, $7$ and $3$, $4$, $11$, $8$.
$(Ⅱ)$ Let $A$ be the set of all sequences $\{a_{n}\}$ that satisfy the recursive relation $a_{n+1}=\frac{1+a_{n}}{1-a_{n}}$. Let $\{b_{n}\}$ and $\{c_{n}\}$ be two elements in $A$, both with $m$ terms. If $b_{1}=2$, $c_{1}=3$, and the distance between $\{b_{n}\}$ and $\{c_{n}\}$ is less than $2016$, find the maximum value of $m$.
$(Ⅲ)$ Let $S$ be the set of all $7$-term sequences $\{a_{n}|1\leqslant n\leqslant 7$, $a_{n}=0$ or $1\}$. Let $T\subseteq S$ such that the distance between any two elements in $T$ is greater than or equal to $3$. Prove that the number of elements in $T$ is less than or equal to $16$.<|im_end|>
<|im_start|>assistant
<Z_480><Z_111><Z_173><Z_288><Z_243><ANSWER>

Question:
Let the number of terms in the sequences $\{a_{n}\}$ and $\{b_{n}\}$ be $m$. The distance between the sequences $\{a_{n}\}$ and $\{b_{n}\}$ is defined as $\sum_{i=1}^{m}|a_{i}-b_{i}|$.
$(Ⅰ)$ Find the distance between the sequences $1$, $4$, $6$, $7$ and $3$, $4$, $11$, $8$.
$(Ⅱ)$ Let $A$ be the set of all sequences $\{a_{n}\}$ that satisfy the recursive relation $a_{n+1}=\frac{1+a_{n}}{1-a_{n}}$. Let $\{b_{n}\}$ and $\{c_{n}\}$ be two elements in $A$, both with $m$ terms. If $b_{1}=2$, $c_{1}=3$, and the distance between $\{b_{n}\}$ and $\{c_{n}\}$ is less than $2016$, find the maximum value of $m$.
$(Ⅲ)$ Let $S$ be the set of all $7$-term sequences $\{a_{n}|1\leqslant n\leqslant 7$, $a_{n}=0$ or $1\}$. Let $T\subseteq S$ such that the distance between any two elements in $T$ is greater than or equal to $3$. Prove that the number of elements in $T$ is less than or equal to $16$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the minimum value of | x - 4 | + | x + 7 | + | x - 5 | ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the minimum value of | x - 4 | + | x + 7 | + | x - 5 | ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_417><Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the minimum value of | x - 4 | + | x + 7 | + | x - 5 | ?<|im_end|>
<|im_start|>assistant
<Z_508><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the minimum value of | x - 4 | + | x + 7 | + | x - 5 | ?<|im_end|>
<|im_start|>assistant
<Z_242><Z_204><Z_504><ANSWER>

Question: what is the minimum value of | x - 4 | + | x + 7 | + | x - 5 | ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike took a taxi to the airport and paid $2.50 to start plus $0.25 per mile. Annie took a different route to the airport and paid $2.50 plus $5.00 in bridge toll fees plus $0.25 per mile. If each was charged exactly the same amount, and Annie's ride was 22 miles, how many miles was Mike's ride?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike took a taxi to the airport and paid $2.50 to start plus $0.25 per mile. Annie took a different route to the airport and paid $2.50 plus $5.00 in bridge toll fees plus $0.25 per mile. If each was charged exactly the same amount, and Annie's ride was 22 miles, how many miles was Mike's ride?<|im_end|>
<|im_start|>assistant
<Z_24><Z_162><Z_343><Z_106><Z_215><Z_60><Z_151><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike took a taxi to the airport and paid $2.50 to start plus $0.25 per mile. Annie took a different route to the airport and paid $2.50 plus $5.00 in bridge toll fees plus $0.25 per mile. If each was charged exactly the same amount, and Annie's ride was 22 miles, how many miles was Mike's ride?<|im_end|>
<|im_start|>assistant
<Z_297><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike took a taxi to the airport and paid $2.50 to start plus $0.25 per mile. Annie took a different route to the airport and paid $2.50 plus $5.00 in bridge toll fees plus $0.25 per mile. If each was charged exactly the same amount, and Annie's ride was 22 miles, how many miles was Mike's ride?<|im_end|>
<|im_start|>assistant
<Z_216><Z_271><Z_203><Z_299><Z_101><Z_179><Z_289><Z_392><ANSWER>

Question: Mike took a taxi to the airport and paid $2.50 to start plus $0.25 per mile. Annie took a different route to the airport and paid $2.50 plus $5.00 in bridge toll fees plus $0.25 per mile. If each was charged exactly the same amount, and Annie's ride was 22 miles, how many miles was Mike's ride?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?<|im_end|>
<|im_start|>assistant
<Z_494><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?<|im_end|>
<|im_start|>assistant
<Z_178><ANSWER>

Question: Summer, Jolly, Winter, and Autumn all went to the same middle school. When they finished their education, the degrees they earned were in the ratio of 5:4:6:3. If Summer, who had the highest number of degrees, has a total of 150 degrees, what's the combined number of degrees they all have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average salary of all the workers in a workshop is Rs. 8000. The average salary of 7 technicians is Rs. 18000 and the average salary of the rest is some amount. The total number of workers in the workshop is 42. What is the average salary of the rest of the workers?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average salary of all the workers in a workshop is Rs. 8000. The average salary of 7 technicians is Rs. 18000 and the average salary of the rest is some amount. The total number of workers in the workshop is 42. What is the average salary of the rest of the workers?<|im_end|>
<|im_start|>assistant
<Z_24><Z_229><Z_99><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average salary of all the workers in a workshop is Rs. 8000. The average salary of 7 technicians is Rs. 18000 and the average salary of the rest is some amount. The total number of workers in the workshop is 42. What is the average salary of the rest of the workers?<|im_end|>
<|im_start|>assistant
<Z_413><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average salary of all the workers in a workshop is Rs. 8000. The average salary of 7 technicians is Rs. 18000 and the average salary of the rest is some amount. The total number of workers in the workshop is 42. What is the average salary of the rest of the workers?<|im_end|>
<|im_start|>assistant
<Z_169><Z_420><Z_262><Z_300><ANSWER>

Question: The average salary of all the workers in a workshop is Rs. 8000. The average salary of 7 technicians is Rs. 18000 and the average salary of the rest is some amount. The total number of workers in the workshop is 42. What is the average salary of the rest of the workers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the following infinite list of numbers, how many are integers? $$\sqrt{2401},\sqrt[3]{2401},\sqrt[4]{2401},\sqrt[5]{2401},\sqrt[6]{2401},\ldots$$<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the following infinite list of numbers, how many are integers? $$\sqrt{2401},\sqrt[3]{2401},\sqrt[4]{2401},\sqrt[5]{2401},\sqrt[6]{2401},\ldots$$<|im_end|>
<|im_start|>assistant
<Z_24><Z_493><Z_236><Z_391><Z_310><Z_35><Z_269><Z_119><Z_329><Z_269><Z_376><Z_287><Z_261><Z_436><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the following infinite list of numbers, how many are integers? $$\sqrt{2401},\sqrt[3]{2401},\sqrt[4]{2401},\sqrt[5]{2401},\sqrt[6]{2401},\ldots$$<|im_end|>
<|im_start|>assistant
<Z_292><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the following infinite list of numbers, how many are integers? $$\sqrt{2401},\sqrt[3]{2401},\sqrt[4]{2401},\sqrt[5]{2401},\sqrt[6]{2401},\ldots$$<|im_end|>
<|im_start|>assistant
<Z_237><Z_20><Z_453><Z_176><Z_20><Z_33><Z_275><Z_323><Z_422><Z_478><Z_296><Z_33><Z_34><Z_90><ANSWER>

Question: From the following infinite list of numbers, how many are integers? $$\sqrt{2401},\sqrt[3]{2401},\sqrt[4]{2401},\sqrt[5]{2401},\sqrt[6]{2401},\ldots$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different three-letter sets of initials are possible using the letters A through J, if the letter J cannot be the first initial?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different three-letter sets of initials are possible using the letters A through J, if the letter J cannot be the first initial?<|im_end|>
<|im_start|>assistant
<Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different three-letter sets of initials are possible using the letters A through J, if the letter J cannot be the first initial?<|im_end|>
<|im_start|>assistant
<Z_171><ANSWER>

Question: How many different three-letter sets of initials are possible using the letters A through J, if the letter J cannot be the first initial?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use $\left[x\right]$ to represent the largest integer not greater than $x$, for example, $\left[2.4\right]=2$, $\left[-3.1\right]=-4$. Please calculate $[{5.5}]+[{-4\frac{1}{2}}]=$____.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use $\left[x\right]$ to represent the largest integer not greater than $x$, for example, $\left[2.4\right]=2$, $\left[-3.1\right]=-4$. Please calculate $[{5.5}]+[{-4\frac{1}{2}}]=$____.<|im_end|>
<|im_start|>assistant
<Z_477><Z_200><Z_140><Z_322><Z_67><Z_61><Z_72><Z_464><Z_470><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use $\left[x\right]$ to represent the largest integer not greater than $x$, for example, $\left[2.4\right]=2$, $\left[-3.1\right]=-4$. Please calculate $[{5.5}]+[{-4\frac{1}{2}}]=$____.<|im_end|>
<|im_start|>assistant
<Z_448><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use $\left[x\right]$ to represent the largest integer not greater than $x$, for example, $\left[2.4\right]=2$, $\left[-3.1\right]=-4$. Please calculate $[{5.5}]+[{-4\frac{1}{2}}]=$____.<|im_end|>
<|im_start|>assistant
<Z_381><Z_281><Z_414><Z_92><Z_483><Z_275><Z_201><Z_422><Z_258><ANSWER>

Question: Use $\left[x\right]$ to represent the largest integer not greater than $x$, for example, $\left[2.4\right]=2$, $\left[-3.1\right]=-4$. Please calculate $[{5.5}]+[{-4\frac{1}{2}}]=$____.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paige's team won their dodgeball game and scored 81 points in total. The team scored through the following methods: hitting opponents with the ball (2 points each), catching opponents' throws (5 points each), and eliminating all players in an opponent's team (10 points). If Paige scored 21 points through a combination of hitting and catching, and her teammate Brian scored 20 points through a combination of catching and eliminating opponents' teams, how many points did each of Karen, Jennifer, and Michael individually score in the game, considering that none of them scored any points by eliminating an opponent's team?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paige's team won their dodgeball game and scored 81 points in total. The team scored through the following methods: hitting opponents with the ball (2 points each), catching opponents' throws (5 points each), and eliminating all players in an opponent's team (10 points). If Paige scored 21 points through a combination of hitting and catching, and her teammate Brian scored 20 points through a combination of catching and eliminating opponents' teams, how many points did each of Karen, Jennifer, and Michael individually score in the game, considering that none of them scored any points by eliminating an opponent's team?<|im_end|>
<|im_start|>assistant
<Z_369><Z_360><Z_263><Z_311><Z_453><Z_230><Z_309><Z_80><Z_246><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paige's team won their dodgeball game and scored 81 points in total. The team scored through the following methods: hitting opponents with the ball (2 points each), catching opponents' throws (5 points each), and eliminating all players in an opponent's team (10 points). If Paige scored 21 points through a combination of hitting and catching, and her teammate Brian scored 20 points through a combination of catching and eliminating opponents' teams, how many points did each of Karen, Jennifer, and Michael individually score in the game, considering that none of them scored any points by eliminating an opponent's team?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paige's team won their dodgeball game and scored 81 points in total. The team scored through the following methods: hitting opponents with the ball (2 points each), catching opponents' throws (5 points each), and eliminating all players in an opponent's team (10 points). If Paige scored 21 points through a combination of hitting and catching, and her teammate Brian scored 20 points through a combination of catching and eliminating opponents' teams, how many points did each of Karen, Jennifer, and Michael individually score in the game, considering that none of them scored any points by eliminating an opponent's team?<|im_end|>
<|im_start|>assistant
<Z_367><Z_147><Z_350><Z_385><Z_491><Z_300><Z_175><Z_189><Z_161><ANSWER>

Question: Paige's team won their dodgeball game and scored 81 points in total. The team scored through the following methods: hitting opponents with the ball (2 points each), catching opponents' throws (5 points each), and eliminating all players in an opponent's team (10 points). If Paige scored 21 points through a combination of hitting and catching, and her teammate Brian scored 20 points through a combination of catching and eliminating opponents' teams, how many points did each of Karen, Jennifer, and Michael individually score in the game, considering that none of them scored any points by eliminating an opponent's team?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive $c$ such that $c$ is a multiple of $6$ and $c$ is a multiple of $15$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive $c$ such that $c$ is a multiple of $6$ and $c$ is a multiple of $15$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_68><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive $c$ such that $c$ is a multiple of $6$ and $c$ is a multiple of $15$.<|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
Find the smallest positive $c$ such that $c$ is a multiple of $6$ and $c$ is a multiple of $15$.<|im_end|>
<|im_start|>assistant
<Z_351><Z_294><Z_422><ANSWER>

Question: Find the smallest positive $c$ such that $c$ is a multiple of $6$ and $c$ is a multiple of $15$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 86 students, each student is registered for at least one of 3 classes – history, math and English. Some students are registered for history, 20 - 3 students are registered for math, and forty - 4 students are registered for English. Only 3 students are registered for all 3 classes. There are 3 students registered for exactly two classes. How many students are registered for history?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 86 students, each student is registered for at least one of 3 classes – history, math and English. Some students are registered for history, 20 - 3 students are registered for math, and forty - 4 students are registered for English. Only 3 students are registered for all 3 classes. There are 3 students registered for exactly two classes. How many students are registered for history?<|im_end|>
<|im_start|>assistant
<Z_477><Z_321><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 86 students, each student is registered for at least one of 3 classes – history, math and English. Some students are registered for history, 20 - 3 students are registered for math, and forty - 4 students are registered for English. Only 3 students are registered for all 3 classes. There are 3 students registered for exactly two classes. How many students are registered for history?<|im_end|>
<|im_start|>assistant
<Z_203><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 86 students, each student is registered for at least one of 3 classes – history, math and English. Some students are registered for history, 20 - 3 students are registered for math, and forty - 4 students are registered for English. Only 3 students are registered for all 3 classes. There are 3 students registered for exactly two classes. How many students are registered for history?<|im_end|>
<|im_start|>assistant
<Z_249><Z_504><Z_299><ANSWER>

Question: In a group of 86 students, each student is registered for at least one of 3 classes – history, math and English. Some students are registered for history, 20 - 3 students are registered for math, and forty - 4 students are registered for English. Only 3 students are registered for all 3 classes. There are 3 students registered for exactly two classes. How many students are registered for history?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, some social studies problems, and 10 science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many social studies problems does he have?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, some social studies problems, and 10 science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many social studies problems does he have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_32><Z_213><Z_57><Z_241><Z_494><Z_211><Z_150><Z_278><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, some social studies problems, and 10 science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many social studies problems does he have?<|im_end|>
<|im_start|>assistant
<Z_398><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brooke has 15 math problems, some social studies problems, and 10 science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many social studies problems does he have?<|im_end|>
<|im_start|>assistant
<Z_68><Z_205><Z_151><Z_491><Z_266><Z_399><Z_197><Z_327><Z_192><Z_495><ANSWER>

Question: Brooke has 15 math problems, some social studies problems, and 10 science problems for homework. He can answer each math problem for 2 minutes, each social studies problem for 30 seconds, and each science problem for 1.5 minutes. It takes Brooke 48 minutes to answer all his homework. How many social studies problems does he have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Triangle $XYZ$ is an isosceles triangle with side lengths of 13, 13, and 24 centimeters. What is the area of triangle $XYZ$, in square centimeters?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Triangle $XYZ$ is an isosceles triangle with side lengths of 13, 13, and 24 centimeters. What is the area of triangle $XYZ$, in square centimeters?<|im_end|>
<|im_start|>assistant
<Z_24><Z_178><Z_211><Z_311><Z_484><Z_506><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Triangle $XYZ$ is an isosceles triangle with side lengths of 13, 13, and 24 centimeters. What is the area of triangle $XYZ$, in square centimeters?<|im_end|>
<|im_start|>assistant
<Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Triangle $XYZ$ is an isosceles triangle with side lengths of 13, 13, and 24 centimeters. What is the area of triangle $XYZ$, in square centimeters?<|im_end|>
<|im_start|>assistant
<Z_343><Z_474><Z_7><Z_279><Z_91><Z_313><ANSWER>

Question: Triangle $XYZ$ is an isosceles triangle with side lengths of 13, 13, and 24 centimeters. What is the area of triangle $XYZ$, in square centimeters?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Garcia family consists of two adults (Mr. and Mrs. Garcia), their two children with the eldest being responsible enough to drive, and their grandmother who does not drive. When driving somewhere together in their car, two people sit in the front including the driver, while the three others sit in the back. The driver must be either Mr. Garcia, Mrs. Garcia, or the eldest child. How many seating arrangements are possible?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Garcia family consists of two adults (Mr. and Mrs. Garcia), their two children with the eldest being responsible enough to drive, and their grandmother who does not drive. When driving somewhere together in their car, two people sit in the front including the driver, while the three others sit in the back. The driver must be either Mr. Garcia, Mrs. Garcia, or the eldest child. How many seating arrangements are possible?<|im_end|>
<|im_start|>assistant
<Z_24><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Garcia family consists of two adults (Mr. and Mrs. Garcia), their two children with the eldest being responsible enough to drive, and their grandmother who does not drive. When driving somewhere together in their car, two people sit in the front including the driver, while the three others sit in the back. The driver must be either Mr. Garcia, Mrs. Garcia, or the eldest child. How many seating arrangements are possible?<|im_end|>
<|im_start|>assistant
<Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Garcia family consists of two adults (Mr. and Mrs. Garcia), their two children with the eldest being responsible enough to drive, and their grandmother who does not drive. When driving somewhere together in their car, two people sit in the front including the driver, while the three others sit in the back. The driver must be either Mr. Garcia, Mrs. Garcia, or the eldest child. How many seating arrangements are possible?<|im_end|>
<|im_start|>assistant
<Z_321><Z_404><ANSWER>

Question: The Garcia family consists of two adults (Mr. and Mrs. Garcia), their two children with the eldest being responsible enough to drive, and their grandmother who does not drive. When driving somewhere together in their car, two people sit in the front including the driver, while the three others sit in the back. The driver must be either Mr. Garcia, Mrs. Garcia, or the eldest child. How many seating arrangements are possible?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?<|im_end|>
<|im_start|>assistant
<Z_24><Z_336><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?<|im_end|>
<|im_start|>assistant
<Z_184><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?<|im_end|>
<|im_start|>assistant
<Z_342><Z_223><ANSWER>

Question: What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The base $8$ representation of a positive integer is $AB$ and its base $5$ representation is $BA$. What is the integer expressed in base $10$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The base $8$ representation of a positive integer is $AB$ and its base $5$ representation is $BA$. What is the integer expressed in base $10$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_331><Z_316><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The base $8$ representation of a positive integer is $AB$ and its base $5$ representation is $BA$. What is the integer expressed in base $10$?<|im_end|>
<|im_start|>assistant
<Z_509><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The base $8$ representation of a positive integer is $AB$ and its base $5$ representation is $BA$. What is the integer expressed in base $10$?<|im_end|>
<|im_start|>assistant
<Z_448><Z_109><Z_384><ANSWER>

Question: The base $8$ representation of a positive integer is $AB$ and its base $5$ representation is $BA$. What is the integer expressed in base $10$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what maximum integer \( n \) do two solutions of the equation \( x^3 - (n+9)x^2 + (2n^2 - 3n - 34)x + 2(n-4)(n+3) = 0 \) exceed 2?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what maximum integer \( n \) do two solutions of the equation \( x^3 - (n+9)x^2 + (2n^2 - 3n - 34)x + 2(n-4)(n+3) = 0 \) exceed 2?<|im_end|>
<|im_start|>assistant
<Z_24><Z_293><Z_248><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what maximum integer \( n \) do two solutions of the equation \( x^3 - (n+9)x^2 + (2n^2 - 3n - 34)x + 2(n-4)(n+3) = 0 \) exceed 2?<|im_end|>
<|im_start|>assistant
<Z_317><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what maximum integer \( n \) do two solutions of the equation \( x^3 - (n+9)x^2 + (2n^2 - 3n - 34)x + 2(n-4)(n+3) = 0 \) exceed 2?<|im_end|>
<|im_start|>assistant
<Z_370><Z_236><Z_426><ANSWER>

Question: For what maximum integer \( n \) do two solutions of the equation \( x^3 - (n+9)x^2 + (2n^2 - 3n - 34)x + 2(n-4)(n+3) = 0 \) exceed 2?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[g(n) =
\begin{cases}
n^2 + 2n - 1 & \text{ if } n < 5,
\\ 2n + 3 & \text{ if } n \geq 5.
\end{cases}
\]Find $g(g(g(3)))$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[g(n) =
\begin{cases}
n^2 + 2n - 1 & \text{ if } n < 5,
\\ 2n + 3 & \text{ if } n \geq 5.
\end{cases}
\]Find $g(g(g(3)))$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_362><Z_443><Z_420><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[g(n) =
\begin{cases}
n^2 + 2n - 1 & \text{ if } n < 5,
\\ 2n + 3 & \text{ if } n \geq 5.
\end{cases}
\]Find $g(g(g(3)))$.<|im_end|>
<|im_start|>assistant
<Z_339><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[g(n) =
\begin{cases}
n^2 + 2n - 1 & \text{ if } n < 5,
\\ 2n + 3 & \text{ if } n \geq 5.
\end{cases}
\]Find $g(g(g(3)))$.<|im_end|>
<|im_start|>assistant
<Z_282><Z_38><Z_114><Z_437><ANSWER>

Question:
Let \[g(n) =
\begin{cases}
n^2 + 2n - 1 & \text{ if } n < 5,
\\ 2n + 3 & \text{ if } n \geq 5.
\end{cases}
\]Find $g(g(g(3)))$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the remainder if 7 ^ 4 is divided by 100 ?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the remainder if 7 ^ 4 is divided by 100 ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_228><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the remainder if 7 ^ 4 is divided by 100 ?<|im_end|>
<|im_start|>assistant
<Z_207><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
what is the remainder if 7 ^ 4 is divided by 100 ?<|im_end|>
<|im_start|>assistant
<Z_288><Z_32><ANSWER>

Question: what is the remainder if 7 ^ 4 is divided by 100 ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five plastic bottles are required to make a new bottle. How many new bottles can eventually be made from 625 plastic bottles, assuming that at least three bottles are needed to start a recycling process? Do not include the original 625 bottles in your count.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five plastic bottles are required to make a new bottle. How many new bottles can eventually be made from 625 plastic bottles, assuming that at least three bottles are needed to start a recycling process? Do not include the original 625 bottles in your count.<|im_end|>
<|im_start|>assistant
<Z_477><Z_449><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five plastic bottles are required to make a new bottle. How many new bottles can eventually be made from 625 plastic bottles, assuming that at least three bottles are needed to start a recycling process? Do not include the original 625 bottles in your count.<|im_end|>
<|im_start|>assistant
<Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five plastic bottles are required to make a new bottle. How many new bottles can eventually be made from 625 plastic bottles, assuming that at least three bottles are needed to start a recycling process? Do not include the original 625 bottles in your count.<|im_end|>
<|im_start|>assistant
<Z_44><Z_161><Z_28><ANSWER>

Question: Five plastic bottles are required to make a new bottle. How many new bottles can eventually be made from 625 plastic bottles, assuming that at least three bottles are needed to start a recycling process? Do not include the original 625 bottles in your count.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train 560 m in length crosses a telegraph post in a certain amount of time. The speed of the train is 126 km/h. How many seconds does it take for the train to cross the telegraph post?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train 560 m in length crosses a telegraph post in a certain amount of time. The speed of the train is 126 km/h. How many seconds does it take for the train to cross the telegraph post?<|im_end|>
<|im_start|>assistant
<Z_369><Z_272><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train 560 m in length crosses a telegraph post in a certain amount of time. The speed of the train is 126 km/h. How many seconds does it take for the train to cross the telegraph post?<|im_end|>
<|im_start|>assistant
<Z_354><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train 560 m in length crosses a telegraph post in a certain amount of time. The speed of the train is 126 km/h. How many seconds does it take for the train to cross the telegraph post?<|im_end|>
<|im_start|>assistant
<Z_390><Z_309><ANSWER>

Question: A train 560 m in length crosses a telegraph post in a certain amount of time. The speed of the train is 126 km/h. How many seconds does it take for the train to cross the telegraph post?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company charges a $\$80$ set-up fee plus $\$10$ for each shirt printed. Beta T-shirt Company charges a $\$20$ set up fee, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company charges a $\$80$ set-up fee plus $\$10$ for each shirt printed. Beta T-shirt Company charges a $\$20$ set up fee, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company charges a $\$80$ set-up fee plus $\$10$ for each shirt printed. Beta T-shirt Company charges a $\$20$ set up fee, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_246><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company charges a $\$80$ set-up fee plus $\$10$ for each shirt printed. Beta T-shirt Company charges a $\$20$ set up fee, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_146><ANSWER>

Question: Acme T-Shirt Company charges a $\$80$ set-up fee plus $\$10$ for each shirt printed. Beta T-shirt Company charges a $\$20$ set up fee, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An escalator moves towards the top level at the rate of 15 ft/sec and its length is 180 feet. If a person walks on the moving escalator at a certain rate towards the top level, he takes 10 seconds to cover the entire length. What is the rate at which the person walks on the escalator?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An escalator moves towards the top level at the rate of 15 ft/sec and its length is 180 feet. If a person walks on the moving escalator at a certain rate towards the top level, he takes 10 seconds to cover the entire length. What is the rate at which the person walks on the escalator?<|im_end|>
<|im_start|>assistant
<Z_477><Z_157><Z_186><Z_343><Z_206><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An escalator moves towards the top level at the rate of 15 ft/sec and its length is 180 feet. If a person walks on the moving escalator at a certain rate towards the top level, he takes 10 seconds to cover the entire length. What is the rate at which the person walks on the escalator?<|im_end|>
<|im_start|>assistant
<Z_488><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An escalator moves towards the top level at the rate of 15 ft/sec and its length is 180 feet. If a person walks on the moving escalator at a certain rate towards the top level, he takes 10 seconds to cover the entire length. What is the rate at which the person walks on the escalator?<|im_end|>
<|im_start|>assistant
<Z_166><Z_154><Z_14><Z_478><Z_425><ANSWER>

Question: An escalator moves towards the top level at the rate of 15 ft/sec and its length is 180 feet. If a person walks on the moving escalator at a certain rate towards the top level, he takes 10 seconds to cover the entire length. What is the rate at which the person walks on the escalator?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To asphalt a 1 km road, 30 men spent 12 days working 8 hours per day. It takes a certain number of men working 9 hours a day 32 days to asphalt a road of 2 km. How many men are working on the second road?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To asphalt a 1 km road, 30 men spent 12 days working 8 hours per day. It takes a certain number of men working 9 hours a day 32 days to asphalt a road of 2 km. How many men are working on the second road?<|im_end|>
<|im_start|>assistant
<Z_24><Z_162><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To asphalt a 1 km road, 30 men spent 12 days working 8 hours per day. It takes a certain number of men working 9 hours a day 32 days to asphalt a road of 2 km. How many men are working on the second road?<|im_end|>
<|im_start|>assistant
<Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To asphalt a 1 km road, 30 men spent 12 days working 8 hours per day. It takes a certain number of men working 9 hours a day 32 days to asphalt a road of 2 km. How many men are working on the second road?<|im_end|>
<|im_start|>assistant
<Z_201><Z_397><ANSWER>

Question: To asphalt a 1 km road, 30 men spent 12 days working 8 hours per day. It takes a certain number of men working 9 hours a day 32 days to asphalt a road of 2 km. How many men are working on the second road?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_272><Z_343><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ?<|im_end|>
<|im_start|>assistant
<Z_420><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ?<|im_end|>
<|im_start|>assistant
<Z_475><Z_367><Z_283><ANSWER>

Question: arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when 2,345,678,901 is divided by 101?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when 2,345,678,901 is divided by 101?<|im_end|>
<|im_start|>assistant
<Z_24><Z_400><Z_413><Z_193><Z_278><Z_467><Z_423><Z_159><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when 2,345,678,901 is divided by 101?<|im_end|>
<|im_start|>assistant
<Z_496><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when 2,345,678,901 is divided by 101?<|im_end|>
<|im_start|>assistant
<Z_157><Z_431><Z_443><Z_187><Z_278><Z_77><Z_170><Z_507><ANSWER>

Question: What is the remainder when 2,345,678,901 is divided by 101?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A bag contains $n$ red balls $(n \geqslant 5$ and $n \in \mathbb{N})$ and $5$ white balls. In a lottery draw, two balls are drawn at once, and it is considered a win if the two balls are of different colors.
$(1)$ Express the probability of winning in a single draw, $p_n$, in terms of $n$;
$(2)$ If $n=5$, and assuming that after each draw the balls are replaced, let $X$ be the number of wins in three draws, find the expected value $EX$ of $X$;
$(3)$ Assuming that after each draw the balls are replaced, let $P$ be the probability of exactly one win in three draws. For what value of $n$ is $P$ maximized?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A bag contains $n$ red balls $(n \geqslant 5$ and $n \in \mathbb{N})$ and $5$ white balls. In a lottery draw, two balls are drawn at once, and it is considered a win if the two balls are of different colors.
$(1)$ Express the probability of winning in a single draw, $p_n$, in terms of $n$;
$(2)$ If $n=5$, and assuming that after each draw the balls are replaced, let $X$ be the number of wins in three draws, find the expected value $EX$ of $X$;
$(3)$ Assuming that after each draw the balls are replaced, let $P$ be the probability of exactly one win in three draws. For what value of $n$ is $P$ maximized?<|im_end|>
<|im_start|>assistant
<Z_477><Z_382><Z_230><Z_52><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A bag contains $n$ red balls $(n \geqslant 5$ and $n \in \mathbb{N})$ and $5$ white balls. In a lottery draw, two balls are drawn at once, and it is considered a win if the two balls are of different colors.
$(1)$ Express the probability of winning in a single draw, $p_n$, in terms of $n$;
$(2)$ If $n=5$, and assuming that after each draw the balls are replaced, let $X$ be the number of wins in three draws, find the expected value $EX$ of $X$;
$(3)$ Assuming that after each draw the balls are replaced, let $P$ be the probability of exactly one win in three draws. For what value of $n$ is $P$ maximized?<|im_end|>
<|im_start|>assistant
<Z_223><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A bag contains $n$ red balls $(n \geqslant 5$ and $n \in \mathbb{N})$ and $5$ white balls. In a lottery draw, two balls are drawn at once, and it is considered a win if the two balls are of different colors.
$(1)$ Express the probability of winning in a single draw, $p_n$, in terms of $n$;
$(2)$ If $n=5$, and assuming that after each draw the balls are replaced, let $X$ be the number of wins in three draws, find the expected value $EX$ of $X$;
$(3)$ Assuming that after each draw the balls are replaced, let $P$ be the probability of exactly one win in three draws. For what value of $n$ is $P$ maximized?<|im_end|>
<|im_start|>assistant
<Z_44><Z_123><Z_14><Z_209><Z_81><ANSWER>

Question:
A bag contains $n$ red balls $(n \geqslant 5$ and $n \in \mathbb{N})$ and $5$ white balls. In a lottery draw, two balls are drawn at once, and it is considered a win if the two balls are of different colors.
$(1)$ Express the probability of winning in a single draw, $p_n$, in terms of $n$;
$(2)$ If $n=5$, and assuming that after each draw the balls are replaced, let $X$ be the number of wins in three draws, find the expected value $EX$ of $X$;
$(3)$ Assuming that after each draw the balls are replaced, let $P$ be the probability of exactly one win in three draws. For what value of $n$ is $P$ maximized?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the least common multiple of $A$ and $B$ is $240$, and the ratio of $A$ to $B$ is $5:6$, then what is their greatest common divisor?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the least common multiple of $A$ and $B$ is $240$, and the ratio of $A$ to $B$ is $5:6$, then what is their greatest common divisor?<|im_end|>
<|im_start|>assistant
<Z_369><Z_255><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the least common multiple of $A$ and $B$ is $240$, and the ratio of $A$ to $B$ is $5:6$, then what is their greatest common divisor?<|im_end|>
<|im_start|>assistant
<Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the least common multiple of $A$ and $B$ is $240$, and the ratio of $A$ to $B$ is $5:6$, then what is their greatest common divisor?<|im_end|>
<|im_start|>assistant
<Z_127><Z_321><ANSWER>

Question: If the least common multiple of $A$ and $B$ is $240$, and the ratio of $A$ to $B$ is $5:6$, then what is their greatest common divisor?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_369><Z_406><Z_467><Z_111><Z_346><Z_150><Z_462><Z_98><Z_202><Z_55><Z_319><Z_155><Z_43><Z_75><Z_246><Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_236><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_386><Z_312><Z_155><Z_111><Z_102><Z_405><Z_281><Z_187><Z_470><Z_445><Z_325><Z_339><Z_442><Z_500><Z_208><Z_112><ANSWER>

Question: Convert the binary number $101101_{(2)}$ to decimal.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream?<|im_end|>
<|im_start|>assistant
<Z_477><Z_124><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream?<|im_end|>
<|im_start|>assistant
<Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream?<|im_end|>
<|im_start|>assistant
<Z_58><Z_131><Z_117><ANSWER>

Question: During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_68><Z_149><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_26><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.<|im_end|>
<|im_start|>assistant
<Z_271><Z_389><Z_379><ANSWER>

Question: Compute: $\displaystyle \frac{125,\!000^5}{25,\!000^5}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_24><Z_437><Z_104><Z_378><Z_146><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_453><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?<|im_end|>
<|im_start|>assistant
<Z_179><Z_61><Z_476><Z_301><Z_342><Z_401><ANSWER>

Question:
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_484><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?<|im_end|>
<|im_start|>assistant
<Z_156><ANSWER>

Question: A train is running at a speed of 40 km/hr and it crosses two consecutive posts in 17.1 seconds and 25.2 seconds respectively. If the distance between the two posts is 190 meters, what is the length of the train?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_477><Z_52><Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_131><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?<|im_end|>
<|im_start|>assistant
<Z_207><Z_390><Z_254><ANSWER>

Question: Tyson played basketball on the schoolyard. He scored three points fifteen times, and two points twelve times. He also scored one point some number of times. How many times did he score one point, if in total he scored 75 points?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_369><Z_43><Z_162><Z_464><Z_110><Z_116><Z_240><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_2><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?<|im_end|>
<|im_start|>assistant
<Z_295><Z_174><Z_374><Z_5><Z_142><Z_52><Z_454><ANSWER>

Question: A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_24><Z_506><Z_206><Z_308><Z_314><Z_211><Z_295><Z_310><Z_429><Z_296><Z_340><Z_296><Z_487><Z_460><Z_391><Z_137><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_151><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?<|im_end|>
<|im_start|>assistant
<Z_349><Z_178><Z_282><Z_49><Z_353><Z_308><Z_416><Z_475><Z_449><Z_286><Z_323><Z_464><Z_54><Z_341><Z_295><Z_457><ANSWER>

Question: Isabelle has two siblings, Anna and Alex, and three cousins, Lily, Jack, and Nick she decided to take to a concert. Isabelle gets a 40% employee discount for her own ticket because she works at the concert venue. Her youngest cousin, Nick, as a child, gets a ticket price at 60% of the standard $80 ticket price. The rest of her siblings and cousins are students, therefore they are entitled to a 20% discount on their tickets. Together, they have collectively saved $150 so far. If Isabelle's part-time job pays her $7.50 per week after her essential weekly expenses, how many more weeks does Isabelle need to work to afford the concert tickets for everyone?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_477><Z_439><Z_419><Z_274><Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_344><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?<|im_end|>
<|im_start|>assistant
<Z_434><Z_23><Z_123><Z_503><Z_489><ANSWER>

Question: Faye had 35 packs of pencils each one having a certain number of pencils. She was placing her pencils into rows with 2 pencils in each row. She could make 70 rows. How many pencils were in each pack?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_369><Z_179><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?<|im_end|>
<|im_start|>assistant
<Z_448><Z_177><ANSWER>

Question: A certain car uses one gallon of gasoline every some miles when it travels on the highway, and one gallon of gasoline every 20 miles when it travels in the city. When a car travels 4 miles on the highway and 4 additional miles in the city, it uses 45.000000000000014% more gasoline than if it travels 8 miles on the highway. How many miles can the car travel on the highway with one gallon of gasoline?