<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_57><Z_27><Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_57><Z_62><Z_30><Z_89><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_58><Z_58><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_60><Z_108><Z_35><Z_89><ANSWER>

Question:
The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_94><Z_99><Z_37><Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?<|im_end|>
<|im_start|>assistant
<Z_53><Z_52><Z_55><Z_78><ANSWER>

Question:
If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?<|im_end|>
<|im_start|>assistant
<Z_115><ANSWER>

Question: Arlene hiked some miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. Arlene's average hiking pace was 4 miles per hour. How many miles did Arlene hike on Saturday?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><Z_28><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?<|im_end|>
<|im_start|>assistant
<Z_47><Z_84><Z_109><Z_91><ANSWER>

Question: Given $a \star b = (a + b)^2 + (a - b)^2$, what is the value of $a \star b$ when $a = 3$ and $b = 5$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_104><Z_68><Z_72><Z_87><Z_72><Z_73><Z_66><Z_91><Z_80><Z_101><Z_60><Z_37><Z_63><Z_120><Z_26><Z_70><Z_117><Z_98><Z_41><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|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
Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?<|im_end|>
<|im_start|>assistant
<Z_4><Z_10><Z_27><Z_122><Z_126><Z_61><Z_37><Z_57><Z_92><Z_27><Z_63><Z_7><Z_126><Z_96><Z_71><Z_0><Z_75><Z_58><Z_1><Z_59><ANSWER>

Question: Tanya tangled the headphone wire and took a picture of the knot, over which she placed a satin ribbon (see figure). How many ways are there to connect the ends of the wire under the ribbon?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that satisfies the congruence \(3x \equiv 15 \pmod{31}\)?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that satisfies the congruence \(3x \equiv 15 \pmod{31}\)?<|im_end|>
<|im_start|>assistant
<Z_104><Z_1><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that satisfies the congruence \(3x \equiv 15 \pmod{31}\)?<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer that satisfies the congruence \(3x \equiv 15 \pmod{31}\)?<|im_end|>
<|im_start|>assistant
<Z_32><Z_43><Z_29><ANSWER>

Question: What is the smallest positive integer that satisfies the congruence \(3x \equiv 15 \pmod{31}\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the quadratic equation \(5x^2 + 20x - 25\). Compute the sum of the squares of its real roots. Express your answer as a common fraction in lowest terms.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the quadratic equation \(5x^2 + 20x - 25\). Compute the sum of the squares of its real roots. Express your answer as a common fraction in lowest terms.<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the quadratic equation \(5x^2 + 20x - 25\). Compute the sum of the squares of its real roots. Express your answer as a common fraction in lowest terms.<|im_end|>
<|im_start|>assistant
<Z_112><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the quadratic equation \(5x^2 + 20x - 25\). Compute the sum of the squares of its real roots. Express your answer as a common fraction in lowest terms.<|im_end|>
<|im_start|>assistant
<Z_63><ANSWER>

Question: Consider the quadratic equation \(5x^2 + 20x - 25\). Compute the sum of the squares of its real roots. Express your answer as a common fraction in lowest terms.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each color copy, print shop X charges $1.25 and print shop Y charges $2.75. The charge for a certain number of color copies at print shop Y is $120 greater than at print shop X. How many color copies were made?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each color copy, print shop X charges $1.25 and print shop Y charges $2.75. The charge for a certain number of color copies at print shop Y is $120 greater than at print shop X. How many color copies were made?<|im_end|>
<|im_start|>assistant
<Z_57><Z_1><Z_37><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each color copy, print shop X charges $1.25 and print shop Y charges $2.75. The charge for a certain number of color copies at print shop Y is $120 greater than at print shop X. How many color copies were made?<|im_end|>
<|im_start|>assistant
<Z_23><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each color copy, print shop X charges $1.25 and print shop Y charges $2.75. The charge for a certain number of color copies at print shop Y is $120 greater than at print shop X. How many color copies were made?<|im_end|>
<|im_start|>assistant
<Z_63><Z_63><Z_36><Z_0><ANSWER>

Question: For each color copy, print shop X charges $1.25 and print shop Y charges $2.75. The charge for a certain number of color copies at print shop Y is $120 greater than at print shop X. How many color copies were made?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cylindrical barrel with a diameter of $10$ feet and a height of $10$ feet is completely filled with water. A solid cube with a side length of $5$ feet is fully submerged and centered horizontally in the barrel. Calculate the volume of water displaced by the cube.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cylindrical barrel with a diameter of $10$ feet and a height of $10$ feet is completely filled with water. A solid cube with a side length of $5$ feet is fully submerged and centered horizontally in the barrel. Calculate the volume of water displaced by the cube.<|im_end|>
<|im_start|>assistant
<Z_104><Z_82><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cylindrical barrel with a diameter of $10$ feet and a height of $10$ feet is completely filled with water. A solid cube with a side length of $5$ feet is fully submerged and centered horizontally in the barrel. Calculate the volume of water displaced by the cube.<|im_end|>
<|im_start|>assistant
<Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cylindrical barrel with a diameter of $10$ feet and a height of $10$ feet is completely filled with water. A solid cube with a side length of $5$ feet is fully submerged and centered horizontally in the barrel. Calculate the volume of water displaced by the cube.<|im_end|>
<|im_start|>assistant
<Z_64><Z_22><Z_126><ANSWER>

Question: A cylindrical barrel with a diameter of $10$ feet and a height of $10$ feet is completely filled with water. A solid cube with a side length of $5$ feet is fully submerged and centered horizontally in the barrel. Calculate the volume of water displaced by the cube.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the geometric sequence $\{a_n\}$, it is known that $a_1 + a_3 = 8$ and $a_5 + a_7 = 4$. Find the value of $a_9 + a_{11} + a_{13} + a_{15}$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the geometric sequence $\{a_n\}$, it is known that $a_1 + a_3 = 8$ and $a_5 + a_7 = 4$. Find the value of $a_9 + a_{11} + a_{13} + a_{15}$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_73><Z_87><Z_114><Z_22><Z_92><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the geometric sequence $\{a_n\}$, it is known that $a_1 + a_3 = 8$ and $a_5 + a_7 = 4$. Find the value of $a_9 + a_{11} + a_{13} + a_{15}$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the geometric sequence $\{a_n\}$, it is known that $a_1 + a_3 = 8$ and $a_5 + a_7 = 4$. Find the value of $a_9 + a_{11} + a_{13} + a_{15}$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_17><Z_23><Z_80><Z_61><Z_40><Z_67><ANSWER>

Question: In the geometric sequence $\{a_n\}$, it is known that $a_1 + a_3 = 8$ and $a_5 + a_7 = 4$. Find the value of $a_9 + a_{11} + a_{13} + a_{15}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a school there are 458 more girls than boys. If there are 692 girls. How many pupils are there in that school?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a school there are 458 more girls than boys. If there are 692 girls. How many pupils are there in that school?<|im_end|>
<|im_start|>assistant
<Z_104><Z_93><Z_47><Z_89><Z_71><Z_81><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a school there are 458 more girls than boys. If there are 692 girls. How many pupils are there in that school?<|im_end|>
<|im_start|>assistant
<Z_25><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a school there are 458 more girls than boys. If there are 692 girls. How many pupils are there in that school?<|im_end|>
<|im_start|>assistant
<Z_72><Z_44><Z_121><Z_19><Z_109><Z_0><Z_60><ANSWER>

Question: In a school there are 458 more girls than boys. If there are 692 girls. How many pupils are there in that school?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jimmy finds that 9 oranges have the same weight as 6 apples. If Jimmy has 45 oranges, how many apples would be required to balance the weight of his 45 oranges?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jimmy finds that 9 oranges have the same weight as 6 apples. If Jimmy has 45 oranges, how many apples would be required to balance the weight of his 45 oranges?<|im_end|>
<|im_start|>assistant
<Z_29><Z_86><Z_105><Z_77><Z_59><Z_34><Z_66><Z_92><Z_58><Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jimmy finds that 9 oranges have the same weight as 6 apples. If Jimmy has 45 oranges, how many apples would be required to balance the weight of his 45 oranges?<|im_end|>
<|im_start|>assistant
<Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jimmy finds that 9 oranges have the same weight as 6 apples. If Jimmy has 45 oranges, how many apples would be required to balance the weight of his 45 oranges?<|im_end|>
<|im_start|>assistant
<Z_84><Z_44><Z_41><Z_32><Z_110><Z_18><Z_35><Z_90><Z_96><Z_82><ANSWER>

Question: Jimmy finds that 9 oranges have the same weight as 6 apples. If Jimmy has 45 oranges, how many apples would be required to balance the weight of his 45 oranges?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of numbers is 7 : 13 and their h . c . f is 23 . their l . c . m is :<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of numbers is 7 : 13 and their h . c . f is 23 . their l . c . m is :<|im_end|>
<|im_start|>assistant
<Z_94><Z_36><Z_119><Z_66><Z_78><Z_98><Z_47><Z_49><Z_63><Z_119><Z_66><Z_119><Z_22><Z_120><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of numbers is 7 : 13 and their h . c . f is 23 . their l . c . m is :<|im_end|>
<|im_start|>assistant
<Z_1><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ratio of numbers is 7 : 13 and their h . c . f is 23 . their l . c . m is :<|im_end|>
<|im_start|>assistant
<Z_33><Z_75><Z_117><Z_22><Z_35><Z_63><Z_95><Z_121><Z_26><Z_39><Z_28><Z_68><Z_104><Z_15><Z_65><ANSWER>

Question: the ratio of numbers is 7 : 13 and their h . c . f is 23 . their l . c . m is :


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows for 5 months, C 35 cows for 4 months, and D 21 cows for 3 months. A's share of rent is Rs. 720. The total rent of the field is Rs. 3250. How many milkmen rented the pasture?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows for 5 months, C 35 cows for 4 months, and D 21 cows for 3 months. A's share of rent is Rs. 720. The total rent of the field is Rs. 3250. How many milkmen rented the pasture?<|im_end|>
<|im_start|>assistant
<Z_94><Z_99><Z_13><Z_72><Z_4><Z_114><Z_114><Z_73><Z_88><Z_41><Z_58><Z_91><Z_58><Z_63><Z_93><Z_48><Z_53><Z_76><Z_101><Z_118><Z_71><Z_117><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows for 5 months, C 35 cows for 4 months, and D 21 cows for 3 months. A's share of rent is Rs. 720. The total rent of the field is Rs. 3250. How many milkmen rented the pasture?<|im_end|>
<|im_start|>assistant
<Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Some milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows for 5 months, C 35 cows for 4 months, and D 21 cows for 3 months. A's share of rent is Rs. 720. The total rent of the field is Rs. 3250. How many milkmen rented the pasture?<|im_end|>
<|im_start|>assistant
<Z_101><Z_110><Z_48><Z_77><Z_89><Z_114><Z_117><Z_53><Z_78><Z_31><Z_54><Z_87><Z_47><Z_54><Z_112><Z_17><Z_55><Z_85><Z_52><Z_122><Z_40><Z_44><Z_97><ANSWER>

Question: Some milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows for 5 months, C 35 cows for 4 months, and D 21 cows for 3 months. A's share of rent is Rs. 720. The total rent of the field is Rs. 3250. How many milkmen rented the pasture?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned was trying to expand his game collection. He bought 50 games from a friend and bought 27 more at a garage sale. If a certain number of games didn't work, and he ended up with 3 good games, how many games didn't work?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned was trying to expand his game collection. He bought 50 games from a friend and bought 27 more at a garage sale. If a certain number of games didn't work, and he ended up with 3 good games, how many games didn't work?<|im_end|>
<|im_start|>assistant
<Z_104><Z_89><Z_72><Z_107><Z_37><Z_70><Z_39><Z_72><Z_39><Z_93><Z_58><Z_5><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned was trying to expand his game collection. He bought 50 games from a friend and bought 27 more at a garage sale. If a certain number of games didn't work, and he ended up with 3 good games, how many games didn't work?<|im_end|>
<|im_start|>assistant
<Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned was trying to expand his game collection. He bought 50 games from a friend and bought 27 more at a garage sale. If a certain number of games didn't work, and he ended up with 3 good games, how many games didn't work?<|im_end|>
<|im_start|>assistant
<Z_25><Z_43><Z_116><Z_123><Z_73><Z_114><Z_77><Z_41><Z_80><Z_78><Z_36><Z_66><Z_101><ANSWER>

Question: Ned was trying to expand his game collection. He bought 50 games from a friend and bought 27 more at a garage sale. If a certain number of games didn't work, and he ended up with 3 good games, how many games didn't work?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest of 6 consecutive integers whose sum is 96?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest of 6 consecutive integers whose sum is 96?<|im_end|>
<|im_start|>assistant
<Z_94><Z_115><Z_107><Z_88><Z_12><Z_91><Z_19><Z_11><Z_66><Z_92><Z_91><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest of 6 consecutive integers whose sum is 96?<|im_end|>
<|im_start|>assistant
<Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest of 6 consecutive integers whose sum is 96?<|im_end|>
<|im_start|>assistant
<Z_84><Z_117><Z_60><Z_88><Z_97><Z_60><Z_46><Z_113><Z_91><Z_75><Z_68><Z_60><ANSWER>

Question: What is the greatest of 6 consecutive integers whose sum is 96?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language now has four words: "splargh," "glumph," "amr," and "blargh." In forming sentences:
- "splargh" cannot come directly before "glumph"
- "amr" cannot come directly before "blargh"
How many valid 3-word sentences can be formed under these restrictions?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language now has four words: "splargh," "glumph," "amr," and "blargh." In forming sentences:
- "splargh" cannot come directly before "glumph"
- "amr" cannot come directly before "blargh"
How many valid 3-word sentences can be formed under these restrictions?<|im_end|>
<|im_start|>assistant
<Z_57><Z_41><Z_99><Z_61><Z_34><Z_56><Z_38><Z_38><Z_48><Z_72><Z_78><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language now has four words: "splargh," "glumph," "amr," and "blargh." In forming sentences:
- "splargh" cannot come directly before "glumph"
- "amr" cannot come directly before "blargh"
How many valid 3-word sentences can be formed under these restrictions?<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Gnollish language now has four words: "splargh," "glumph," "amr," and "blargh." In forming sentences:
- "splargh" cannot come directly before "glumph"
- "amr" cannot come directly before "blargh"
How many valid 3-word sentences can be formed under these restrictions?<|im_end|>
<|im_start|>assistant
<Z_122><Z_20><Z_53><Z_88><Z_34><Z_91><Z_6><Z_103><Z_107><Z_10><Z_28><Z_24><ANSWER>

Question:
The Gnollish language now has four words: "splargh," "glumph," "amr," and "blargh." In forming sentences:
- "splargh" cannot come directly before "glumph"
- "amr" cannot come directly before "blargh"
How many valid 3-word sentences can be formed under these restrictions?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Call a set of positive integers good if there is a partition of it into two sets \( S \) and \( T \), such that there do not exist three elements \( a, b, c \in S \) such that \( a^b = c \) and such that there do not exist three elements \( a, b, c \in T \) such that \( a^b = c \) (with \( a \) and \( b \) not necessarily distinct). Find the smallest positive integer \( n \) such that the set \(\{2,3,4, \ldots, n\}\) is not good.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Call a set of positive integers good if there is a partition of it into two sets \( S \) and \( T \), such that there do not exist three elements \( a, b, c \in S \) such that \( a^b = c \) and such that there do not exist three elements \( a, b, c \in T \) such that \( a^b = c \) (with \( a \) and \( b \) not necessarily distinct). Find the smallest positive integer \( n \) such that the set \(\{2,3,4, \ldots, n\}\) is not good.<|im_end|>
<|im_start|>assistant
<Z_94><Z_1><Z_36><Z_1><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Call a set of positive integers good if there is a partition of it into two sets \( S \) and \( T \), such that there do not exist three elements \( a, b, c \in S \) such that \( a^b = c \) and such that there do not exist three elements \( a, b, c \in T \) such that \( a^b = c \) (with \( a \) and \( b \) not necessarily distinct). Find the smallest positive integer \( n \) such that the set \(\{2,3,4, \ldots, n\}\) is not good.<|im_end|>
<|im_start|>assistant
<Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Call a set of positive integers good if there is a partition of it into two sets \( S \) and \( T \), such that there do not exist three elements \( a, b, c \in S \) such that \( a^b = c \) and such that there do not exist three elements \( a, b, c \in T \) such that \( a^b = c \) (with \( a \) and \( b \) not necessarily distinct). Find the smallest positive integer \( n \) such that the set \(\{2,3,4, \ldots, n\}\) is not good.<|im_end|>
<|im_start|>assistant
<Z_43><Z_105><Z_61><Z_105><ANSWER>

Question: Call a set of positive integers good if there is a partition of it into two sets \( S \) and \( T \), such that there do not exist three elements \( a, b, c \in S \) such that \( a^b = c \) and such that there do not exist three elements \( a, b, c \in T \) such that \( a^b = c \) (with \( a \) and \( b \) not necessarily distinct). Find the smallest positive integer \( n \) such that the set \(\{2,3,4, \ldots, n\}\) is not good.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify $\sqrt{3\cdot 5}\cdot \sqrt{5^3\cdot 3^3}$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify $\sqrt{3\cdot 5}\cdot \sqrt{5^3\cdot 3^3}$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_115><Z_119><Z_31><Z_92><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify $\sqrt{3\cdot 5}\cdot \sqrt{5^3\cdot 3^3}$.<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify $\sqrt{3\cdot 5}\cdot \sqrt{5^3\cdot 3^3}$.<|im_end|>
<|im_start|>assistant
<Z_50><Z_58><Z_19><Z_60><Z_21><Z_99><ANSWER>

Question: Simplify $\sqrt{3\cdot 5}\cdot \sqrt{5^3\cdot 3^3}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression \(x^2 + 16x + 63\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 6x - 72\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression \(x^2 + 16x + 63\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 6x - 72\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?<|im_end|>
<|im_start|>assistant
<Z_94><Z_58><Z_43><Z_97><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression \(x^2 + 16x + 63\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 6x - 72\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?<|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
The expression \(x^2 + 16x + 63\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 6x - 72\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?<|im_end|>
<|im_start|>assistant
<Z_102><Z_24><Z_111><Z_35><ANSWER>

Question: The expression \(x^2 + 16x + 63\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 6x - 72\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that if a number is divisible by 99, then the sum of its digits is at least 18.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that if a number is divisible by 99, then the sum of its digits is at least 18.<|im_end|>
<|im_start|>assistant
<Z_104><Z_82><Z_79><Z_31><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that if a number is divisible by 99, then the sum of its digits is at least 18.<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that if a number is divisible by 99, then the sum of its digits is at least 18.<|im_end|>
<|im_start|>assistant
<Z_21><Z_98><Z_116><Z_20><ANSWER>

Question: Prove that if a number is divisible by 99, then the sum of its digits is at least 18.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base $b$ in which the equation $245_{b} + 132_{b} = 400_{b}$ holds true.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base $b$ in which the equation $245_{b} + 132_{b} = 400_{b}$ holds true.<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><Z_80><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base $b$ in which the equation $245_{b} + 132_{b} = 400_{b}$ holds true.<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base $b$ in which the equation $245_{b} + 132_{b} = 400_{b}$ holds true.<|im_end|>
<|im_start|>assistant
<Z_80><Z_23><Z_37><Z_127><ANSWER>

Question: Find the base $b$ in which the equation $245_{b} + 132_{b} = 400_{b}$ holds true.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = 3x^2 - 2x + 1 \), what is \( f(f(f(1))) \)?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = 3x^2 - 2x + 1 \), what is \( f(f(f(1))) \)?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = 3x^2 - 2x + 1 \), what is \( f(f(f(1))) \)?<|im_end|>
<|im_start|>assistant
<Z_40><ANSWER>

Question: If \( f(x) = 3x^2 - 2x + 1 \), what is \( f(f(f(1))) \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what value of $n$ is the five-digit number $\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what value of $n$ is the five-digit number $\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on).<|im_end|>
<|im_start|>assistant
<Z_57><Z_1><Z_66><Z_93><Z_100><Z_62><Z_73><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what value of $n$ is the five-digit number $\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on).<|im_end|>
<|im_start|>assistant
<Z_25><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For what value of $n$ is the five-digit number $\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on).<|im_end|>
<|im_start|>assistant
<Z_122><Z_73><Z_96><Z_57><Z_28><Z_86><Z_71><Z_56><ANSWER>

Question: For what value of $n$ is the five-digit number $\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.<|im_end|>
<|im_start|>assistant
<Z_104><Z_50><Z_51><Z_34><Z_47><Z_54><Z_11><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.<|im_end|>
<|im_start|>assistant
<Z_1><Z_59><Z_49><Z_93><Z_58><Z_66><Z_78><Z_7><ANSWER>

Question: If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sarah's Shipping and Ryan's Mail Services both ship packages. Sarah's trucks will only carry loads of 18 packages. In contrast, Ryan's trucks will only carry loads of some number of packages. Both businesses ended up shipping the same number of packages this morning, which is 198. How many packages can Ryan's trucks carry in one load?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sarah's Shipping and Ryan's Mail Services both ship packages. Sarah's trucks will only carry loads of 18 packages. In contrast, Ryan's trucks will only carry loads of some number of packages. Both businesses ended up shipping the same number of packages this morning, which is 198. How many packages can Ryan's trucks carry in one load?<|im_end|>
<|im_start|>assistant
<Z_94><Z_121><Z_103><Z_100><Z_28><Z_75><Z_41><Z_28><Z_93><Z_101><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sarah's Shipping and Ryan's Mail Services both ship packages. Sarah's trucks will only carry loads of 18 packages. In contrast, Ryan's trucks will only carry loads of some number of packages. Both businesses ended up shipping the same number of packages this morning, which is 198. How many packages can Ryan's trucks carry in one load?<|im_end|>
<|im_start|>assistant
<Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sarah's Shipping and Ryan's Mail Services both ship packages. Sarah's trucks will only carry loads of 18 packages. In contrast, Ryan's trucks will only carry loads of some number of packages. Both businesses ended up shipping the same number of packages this morning, which is 198. How many packages can Ryan's trucks carry in one load?<|im_end|>
<|im_start|>assistant
<Z_25><Z_15><Z_77><Z_24><Z_51><Z_59><Z_53><Z_93><Z_100><Z_66><Z_40><ANSWER>

Question: Sarah's Shipping and Ryan's Mail Services both ship packages. Sarah's trucks will only carry loads of 18 packages. In contrast, Ryan's trucks will only carry loads of some number of packages. Both businesses ended up shipping the same number of packages this morning, which is 198. How many packages can Ryan's trucks carry in one load?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 2016 points in the plane, with no three points being collinear, demonstrate that it is possible to construct 504 pairwise disjoint quadrilaterals, not necessarily convex, whose vertices are the given points.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 2016 points in the plane, with no three points being collinear, demonstrate that it is possible to construct 504 pairwise disjoint quadrilaterals, not necessarily convex, whose vertices are the given points.<|im_end|>
<|im_start|>assistant
<Z_57><Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 2016 points in the plane, with no three points being collinear, demonstrate that it is possible to construct 504 pairwise disjoint quadrilaterals, not necessarily convex, whose vertices are the given points.<|im_end|>
<|im_start|>assistant
<Z_89><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 2016 points in the plane, with no three points being collinear, demonstrate that it is possible to construct 504 pairwise disjoint quadrilaterals, not necessarily convex, whose vertices are the given points.<|im_end|>
<|im_start|>assistant
<Z_58><Z_7><ANSWER>

Question: Given 2016 points in the plane, with no three points being collinear, demonstrate that it is possible to construct 504 pairwise disjoint quadrilaterals, not necessarily convex, whose vertices are the given points.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the equation $5x + 195y = 975$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the equation $5x + 195y = 975$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?<|im_end|>
<|im_start|>assistant
<Z_94><Z_52><Z_34><Z_124><Z_79><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the equation $5x + 195y = 975$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?<|im_end|>
<|im_start|>assistant
<Z_84><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the equation $5x + 195y = 975$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?<|im_end|>
<|im_start|>assistant
<Z_41><Z_68><Z_87><Z_118><Z_50><Z_17><ANSWER>

Question: The graph of the equation $5x + 195y = 975$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial of degree 8. What is the minimum number of odd numbers that can be in the set \( A_{P} \), given that the number 8 is included in it?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial of degree 8. What is the minimum number of odd numbers that can be in the set \( A_{P} \), given that the number 8 is included in it?<|im_end|>
<|im_start|>assistant
<Z_113><Z_72><Z_36><Z_107><Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial of degree 8. What is the minimum number of odd numbers that can be in the set \( A_{P} \), given that the number 8 is included in it?<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial of degree 8. What is the minimum number of odd numbers that can be in the set \( A_{P} \), given that the number 8 is included in it?<|im_end|>
<|im_start|>assistant
<Z_84><Z_20><Z_40><Z_111><Z_10><ANSWER>

Question: Let \( P \) be a polynomial of degree 8. What is the minimum number of odd numbers that can be in the set \( A_{P} \), given that the number 8 is included in it?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a graph  $G$  on  $n$  vertices, let  $P_G(x)$  be the unique polynomial of degree at most  $n$  such that for each  $i=0,1,2,\dots,n$ ,  $P_G (i)$  equals the number of ways to color the vertices of the graph  $G$  with  $i$  distinct colors such that no two vertices connected by an edge have the same color. For each integer  $3\le k \le 2017$ , define a  $k$ -*tasty* graph to be a connected graph on  $2017$  vertices with  $2017$  edges and a cycle of length  $k$ . Let the *tastiness* of a  $k$ -tasty graph  $G$  be the number of coefficients in  $P_G(x)$  that are odd integers, and let  $t$  be the minimal tastiness over all  $k$ -tasty graphs with  $3\le k \le 2017$ . Determine the sum of all integers  $b$  between  $3$  and  $2017$  inclusive for which there exists a  $b$ -tasty graph with tastiness  $t$ .

*Proposed by Vincent Huang*<|im_end|>
<|im_start|>assistant
<Z_104><Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a graph  $G$  on  $n$  vertices, let  $P_G(x)$  be the unique polynomial of degree at most  $n$  such that for each  $i=0,1,2,\dots,n$ ,  $P_G (i)$  equals the number of ways to color the vertices of the graph  $G$  with  $i$  distinct colors such that no two vertices connected by an edge have the same color. For each integer  $3\le k \le 2017$ , define a  $k$ -*tasty* graph to be a connected graph on  $2017$  vertices with  $2017$  edges and a cycle of length  $k$ . Let the *tastiness* of a  $k$ -tasty graph  $G$  be the number of coefficients in  $P_G(x)$  that are odd integers, and let  $t$  be the minimal tastiness over all  $k$ -tasty graphs with  $3\le k \le 2017$ . Determine the sum of all integers  $b$  between  $3$  and  $2017$  inclusive for which there exists a  $b$ -tasty graph with tastiness  $t$ .

*Proposed by Vincent Huang*<|im_end|>
<|im_start|>assistant
<Z_104><Z_51><Z_51><Z_13><Z_82><Z_114><Z_31><Z_54><Z_56><Z_39><Z_37><Z_38><Z_96><Z_60><Z_61><Z_47><Z_91><Z_66><Z_93><Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a graph  $G$  on  $n$  vertices, let  $P_G(x)$  be the unique polynomial of degree at most  $n$  such that for each  $i=0,1,2,\dots,n$ ,  $P_G (i)$  equals the number of ways to color the vertices of the graph  $G$  with  $i$  distinct colors such that no two vertices connected by an edge have the same color. For each integer  $3\le k \le 2017$ , define a  $k$ -*tasty* graph to be a connected graph on  $2017$  vertices with  $2017$  edges and a cycle of length  $k$ . Let the *tastiness* of a  $k$ -tasty graph  $G$  be the number of coefficients in  $P_G(x)$  that are odd integers, and let  $t$  be the minimal tastiness over all  $k$ -tasty graphs with  $3\le k \le 2017$ . Determine the sum of all integers  $b$  between  $3$  and  $2017$  inclusive for which there exists a  $b$ -tasty graph with tastiness  $t$ .

*Proposed by Vincent Huang*<|im_end|>
<|im_start|>assistant
<Z_74><Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a graph  $G$  on  $n$  vertices, let  $P_G(x)$  be the unique polynomial of degree at most  $n$  such that for each  $i=0,1,2,\dots,n$ ,  $P_G (i)$  equals the number of ways to color the vertices of the graph  $G$  with  $i$  distinct colors such that no two vertices connected by an edge have the same color. For each integer  $3\le k \le 2017$ , define a  $k$ -*tasty* graph to be a connected graph on  $2017$  vertices with  $2017$  edges and a cycle of length  $k$ . Let the *tastiness* of a  $k$ -tasty graph  $G$  be the number of coefficients in  $P_G(x)$  that are odd integers, and let  $t$  be the minimal tastiness over all  $k$ -tasty graphs with  $3\le k \le 2017$ . Determine the sum of all integers  $b$  between  $3$  and  $2017$  inclusive for which there exists a  $b$ -tasty graph with tastiness  $t$ .

*Proposed by Vincent Huang*<|im_end|>
<|im_start|>assistant
<Z_12><Z_118><Z_54><Z_13><Z_68><Z_121><Z_83><Z_110><Z_98><Z_73><Z_52><Z_36><Z_70><Z_100><Z_84><Z_118><Z_34><Z_61><Z_117><Z_24><ANSWER>

Question:
For a graph  $G$  on  $n$  vertices, let  $P_G(x)$  be the unique polynomial of degree at most  $n$  such that for each  $i=0,1,2,\dots,n$ ,  $P_G (i)$  equals the number of ways to color the vertices of the graph  $G$  with  $i$  distinct colors such that no two vertices connected by an edge have the same color. For each integer  $3\le k \le 2017$ , define a  $k$ -*tasty* graph to be a connected graph on  $2017$  vertices with  $2017$  edges and a cycle of length  $k$ . Let the *tastiness* of a  $k$ -tasty graph  $G$  be the number of coefficients in  $P_G(x)$  that are odd integers, and let  $t$  be the minimal tastiness over all  $k$ -tasty graphs with  $3\le k \le 2017$ . Determine the sum of all integers  $b$  between  $3$  and  $2017$  inclusive for which there exists a  $b$ -tasty graph with tastiness  $t$ .

*Proposed by Vincent Huang*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the minimum number of equilateral triangles, each with a side length of 1 unit, needed to cover an equilateral triangle of side length 15 units?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the minimum number of equilateral triangles, each with a side length of 1 unit, needed to cover an equilateral triangle of side length 15 units?<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the minimum number of equilateral triangles, each with a side length of 1 unit, needed to cover an equilateral triangle of side length 15 units?<|im_end|>
<|im_start|>assistant
<Z_35><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the minimum number of equilateral triangles, each with a side length of 1 unit, needed to cover an equilateral triangle of side length 15 units?<|im_end|>
<|im_start|>assistant
<Z_26><ANSWER>

Question: What is the minimum number of equilateral triangles, each with a side length of 1 unit, needed to cover an equilateral triangle of side length 15 units?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 15 days and B in some days. If they work on it together for 8 days, then the fraction of the work that is left is 0.06666666666666665. How many days does it take for B to do the work alone?<|im_end|>
<|im_start|>assistant
<Z_104><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 15 days and B in some days. If they work on it together for 8 days, then the fraction of the work that is left is 0.06666666666666665. How many days does it take for B to do the work alone?<|im_end|>
<|im_start|>assistant
<Z_57><Z_33><Z_87><Z_63><Z_58><Z_34><Z_55><Z_47><Z_66><Z_79><Z_101><Z_101><Z_45><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 15 days and B in some days. If they work on it together for 8 days, then the fraction of the work that is left is 0.06666666666666665. How many days does it take for B to do the work alone?<|im_end|>
<|im_start|>assistant
<Z_52><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 15 days and B in some days. If they work on it together for 8 days, then the fraction of the work that is left is 0.06666666666666665. How many days does it take for B to do the work alone?<|im_end|>
<|im_start|>assistant
<Z_87><Z_3><Z_62><Z_115><Z_11><Z_54><Z_23><Z_99><Z_60><Z_58><Z_96><Z_112><Z_5><Z_88><ANSWER>

Question: A can do a work in 15 days and B in some days. If they work on it together for 8 days, then the fraction of the work that is left is 0.06666666666666665. How many days does it take for B to do the work alone?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Phil likes to collect baseball cards.  He buys a pack of twenty each week for a year, but then loses half of them one day in a fire.  How many baseball cards does Phil have left?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Phil likes to collect baseball cards.  He buys a pack of twenty each week for a year, but then loses half of them one day in a fire.  How many baseball cards does Phil have left?<|im_end|>
<|im_start|>assistant
<Z_57><Z_65><Z_107><Z_34><Z_64><Z_101><Z_80><Z_39><Z_95><Z_33><Z_65><Z_121><Z_82><Z_121><Z_91><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Phil likes to collect baseball cards.  He buys a pack of twenty each week for a year, but then loses half of them one day in a fire.  How many baseball cards does Phil have left?<|im_end|>
<|im_start|>assistant
<Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Phil likes to collect baseball cards.  He buys a pack of twenty each week for a year, but then loses half of them one day in a fire.  How many baseball cards does Phil have left?<|im_end|>
<|im_start|>assistant
<Z_121><Z_17><Z_52><Z_46><Z_41><Z_41><Z_95><Z_91><Z_20><Z_33><Z_107><Z_52><Z_41><Z_19><Z_120><Z_53><ANSWER>

Question: Phil likes to collect baseball cards.  He buys a pack of twenty each week for a year, but then loses half of them one day in a fire.  How many baseball cards does Phil have left?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ and $c$ be nonzero complex numbers such that $a + b + c = 15$ and
\[(a - b)^2 + (a - c)^2 + (b - c)^2 = 2abc.\]Find $\frac{a^3 + b^3 + c^3}{abc}$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ and $c$ be nonzero complex numbers such that $a + b + c = 15$ and
\[(a - b)^2 + (a - c)^2 + (b - c)^2 = 2abc.\]Find $\frac{a^3 + b^3 + c^3}{abc}$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_114><Z_41><Z_121><Z_124><Z_39><Z_9><Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ and $c$ be nonzero complex numbers such that $a + b + c = 15$ and
\[(a - b)^2 + (a - c)^2 + (b - c)^2 = 2abc.\]Find $\frac{a^3 + b^3 + c^3}{abc}$.<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ and $c$ be nonzero complex numbers such that $a + b + c = 15$ and
\[(a - b)^2 + (a - c)^2 + (b - c)^2 = 2abc.\]Find $\frac{a^3 + b^3 + c^3}{abc}$.<|im_end|>
<|im_start|>assistant
<Z_76><Z_28><Z_85><Z_124><Z_98><Z_8><Z_34><Z_84><ANSWER>

Question:
Let $a,$ $b,$ and $c$ be nonzero complex numbers such that $a + b + c = 15$ and
\[(a - b)^2 + (a - c)^2 + (b - c)^2 = 2abc.\]Find $\frac{a^3 + b^3 + c^3}{abc}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackie's favorite cosmetic company was offering free shipping when you spent $50.00. Jackie ordered her favorite shampoo and conditioner that each cost $10.00 a bottle and some bottles of lotion that cost $6.00 each. Jackie needs to spend $12 more to be eligible for free shipping. How many bottles of lotion did Jackie order?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackie's favorite cosmetic company was offering free shipping when you spent $50.00. Jackie ordered her favorite shampoo and conditioner that each cost $10.00 a bottle and some bottles of lotion that cost $6.00 each. Jackie needs to spend $12 more to be eligible for free shipping. How many bottles of lotion did Jackie order?<|im_end|>
<|im_start|>assistant
<Z_27><Z_50><Z_51><Z_124><Z_72><Z_54><Z_78><Z_70><Z_56><Z_67><Z_65><Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackie's favorite cosmetic company was offering free shipping when you spent $50.00. Jackie ordered her favorite shampoo and conditioner that each cost $10.00 a bottle and some bottles of lotion that cost $6.00 each. Jackie needs to spend $12 more to be eligible for free shipping. How many bottles of lotion did Jackie order?<|im_end|>
<|im_start|>assistant
<Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackie's favorite cosmetic company was offering free shipping when you spent $50.00. Jackie ordered her favorite shampoo and conditioner that each cost $10.00 a bottle and some bottles of lotion that cost $6.00 each. Jackie needs to spend $12 more to be eligible for free shipping. How many bottles of lotion did Jackie order?<|im_end|>
<|im_start|>assistant
<Z_57><Z_65><Z_51><Z_3><Z_20><Z_95><Z_78><Z_36><Z_55><Z_4><Z_11><Z_109><ANSWER>

Question: Jackie's favorite cosmetic company was offering free shipping when you spent $50.00. Jackie ordered her favorite shampoo and conditioner that each cost $10.00 a bottle and some bottles of lotion that cost $6.00 each. Jackie needs to spend $12 more to be eligible for free shipping. How many bottles of lotion did Jackie order?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bart buys $10 of notepads for $1.25 each.  They have 60 pages each.  How many pages did he buy?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bart buys $10 of notepads for $1.25 each.  They have 60 pages each.  How many pages did he buy?<|im_end|>
<|im_start|>assistant
<Z_104><Z_59><Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bart buys $10 of notepads for $1.25 each.  They have 60 pages each.  How many pages did he buy?<|im_end|>
<|im_start|>assistant
<Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bart buys $10 of notepads for $1.25 each.  They have 60 pages each.  How many pages did he buy?<|im_end|>
<|im_start|>assistant
<Z_41><Z_7><Z_4><ANSWER>

Question: Bart buys $10 of notepads for $1.25 each.  They have 60 pages each.  How many pages did he buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). \( x \) must also be a power of a prime number. What is the greatest possible value of \( x \)?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). \( x \) must also be a power of a prime number. What is the greatest possible value of \( x \)?<|im_end|>
<|im_start|>assistant
<Z_57><Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). \( x \) must also be a power of a prime number. What is the greatest possible value of \( x \)?<|im_end|>
<|im_start|>assistant
<Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). \( x \) must also be a power of a prime number. What is the greatest possible value of \( x \)?<|im_end|>
<|im_start|>assistant
<Z_5><Z_68><ANSWER>

Question: The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). \( x \) must also be a power of a prime number. What is the greatest possible value of \( x \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Shanghai Cooperation Organization summit will be held in Qingdao in June 2018. The organizing committee plans to allocate five staff members, A, B, C, D, and E, to two different locations to participate in reception work during the conference. If A and B must be in the same group, and each group must have at least 2 people, then the number of different allocation methods is ______.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Shanghai Cooperation Organization summit will be held in Qingdao in June 2018. The organizing committee plans to allocate five staff members, A, B, C, D, and E, to two different locations to participate in reception work during the conference. If A and B must be in the same group, and each group must have at least 2 people, then the number of different allocation methods is ______.<|im_end|>
<|im_start|>assistant
<Z_94><Z_115><Z_80><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Shanghai Cooperation Organization summit will be held in Qingdao in June 2018. The organizing committee plans to allocate five staff members, A, B, C, D, and E, to two different locations to participate in reception work during the conference. If A and B must be in the same group, and each group must have at least 2 people, then the number of different allocation methods is ______.<|im_end|>
<|im_start|>assistant
<Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Shanghai Cooperation Organization summit will be held in Qingdao in June 2018. The organizing committee plans to allocate five staff members, A, B, C, D, and E, to two different locations to participate in reception work during the conference. If A and B must be in the same group, and each group must have at least 2 people, then the number of different allocation methods is ______.<|im_end|>
<|im_start|>assistant
<Z_39><Z_74><Z_115><Z_97><ANSWER>

Question: The Shanghai Cooperation Organization summit will be held in Qingdao in June 2018. The organizing committee plans to allocate five staff members, A, B, C, D, and E, to two different locations to participate in reception work during the conference. If A and B must be in the same group, and each group must have at least 2 people, then the number of different allocation methods is ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lizette scored 92 on her third quiz giving her an average of 94. What is Lizette's average in her first two quizzes?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lizette scored 92 on her third quiz giving her an average of 94. What is Lizette's average in her first two quizzes?<|im_end|>
<|im_start|>assistant
<Z_94><Z_106><Z_12><Z_34><Z_33><Z_53><Z_63><Z_63><Z_66><Z_52><Z_34><Z_90><Z_6><Z_53><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lizette scored 92 on her third quiz giving her an average of 94. What is Lizette's average in her first two quizzes?<|im_end|>
<|im_start|>assistant
<Z_89><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Lizette scored 92 on her third quiz giving her an average of 94. What is Lizette's average in her first two quizzes?<|im_end|>
<|im_start|>assistant
<Z_19><Z_34><Z_47><Z_120><Z_1><Z_120><Z_30><Z_26><Z_23><Z_17><Z_107><Z_39><Z_97><Z_88><Z_23><ANSWER>

Question: Lizette scored 92 on her third quiz giving her an average of 94. What is Lizette's average in her first two quizzes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express the product of $1001_2$ and $121_3$ in base 10.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express the product of $1001_2$ and $121_3$ in base 10.<|im_end|>
<|im_start|>assistant
<Z_104><Z_81><Z_124><Z_31><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express the product of $1001_2$ and $121_3$ in base 10.<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express the product of $1001_2$ and $121_3$ in base 10.<|im_end|>
<|im_start|>assistant
<Z_70><Z_19><Z_64><Z_118><Z_100><ANSWER>

Question: Express the product of $1001_2$ and $121_3$ in base 10.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a fuel station, the service costs $2.20 per vehicle and every liter of fuel costs $0.70. Assuming that you fill up 3 mini-vans and 2 trucks, the total cost is $347.7, if a mini-van's tank is some liters and a truck's tank is 120% bigger and they are all empty. What is the capacity of a mini-van's tank in liters?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a fuel station, the service costs $2.20 per vehicle and every liter of fuel costs $0.70. Assuming that you fill up 3 mini-vans and 2 trucks, the total cost is $347.7, if a mini-van's tank is some liters and a truck's tank is 120% bigger and they are all empty. What is the capacity of a mini-van's tank in liters?<|im_end|>
<|im_start|>assistant
<Z_94><Z_65><Z_108><Z_80><Z_68><Z_12><Z_70><Z_33><Z_87><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a fuel station, the service costs $2.20 per vehicle and every liter of fuel costs $0.70. Assuming that you fill up 3 mini-vans and 2 trucks, the total cost is $347.7, if a mini-van's tank is some liters and a truck's tank is 120% bigger and they are all empty. What is the capacity of a mini-van's tank in liters?<|im_end|>
<|im_start|>assistant
<Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a fuel station, the service costs $2.20 per vehicle and every liter of fuel costs $0.70. Assuming that you fill up 3 mini-vans and 2 trucks, the total cost is $347.7, if a mini-van's tank is some liters and a truck's tank is 120% bigger and they are all empty. What is the capacity of a mini-van's tank in liters?<|im_end|>
<|im_start|>assistant
<Z_106><Z_86><Z_87><Z_111><Z_40><Z_108><Z_4><Z_121><Z_13><Z_9><ANSWER>

Question: In a fuel station, the service costs $2.20 per vehicle and every liter of fuel costs $0.70. Assuming that you fill up 3 mini-vans and 2 trucks, the total cost is $347.7, if a mini-van's tank is some liters and a truck's tank is 120% bigger and they are all empty. What is the capacity of a mini-van's tank in liters?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Luke wants to fence a rectangular piece of land with a total area of at least 600 square feet. Additionally, he requires one side of the rectangle to be exactly 5 feet longer than the other. What should the dimensions of the rectangle be to use the least amount of fencing?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Luke wants to fence a rectangular piece of land with a total area of at least 600 square feet. Additionally, he requires one side of the rectangle to be exactly 5 feet longer than the other. What should the dimensions of the rectangle be to use the least amount of fencing?<|im_end|>
<|im_start|>assistant
<Z_104><Z_79><Z_105><Z_98><Z_36><Z_22><Z_45><Z_45><Z_48><Z_91><Z_82><Z_125><Z_58><Z_70><Z_83><Z_37><Z_75><Z_37><Z_9><Z_79><Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Luke wants to fence a rectangular piece of land with a total area of at least 600 square feet. Additionally, he requires one side of the rectangle to be exactly 5 feet longer than the other. What should the dimensions of the rectangle be to use the least amount of fencing?<|im_end|>
<|im_start|>assistant
<Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Luke wants to fence a rectangular piece of land with a total area of at least 600 square feet. Additionally, he requires one side of the rectangle to be exactly 5 feet longer than the other. What should the dimensions of the rectangle be to use the least amount of fencing?<|im_end|>
<|im_start|>assistant
<Z_104><Z_112><Z_120><Z_52><Z_62><Z_104><Z_60><Z_68><Z_7><Z_11><Z_102><Z_11><Z_47><Z_9><Z_41><Z_88><Z_101><Z_76><Z_119><Z_0><Z_74><ANSWER>

Question: Luke wants to fence a rectangular piece of land with a total area of at least 600 square feet. Additionally, he requires one side of the rectangle to be exactly 5 feet longer than the other. What should the dimensions of the rectangle be to use the least amount of fencing?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)= \begin{cases}\sqrt {x},x > 0 \\ 2^{-x},x\leqslant 0\end{cases}$, find $f[f(-4)]=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)= \begin{cases}\sqrt {x},x > 0 \\ 2^{-x},x\leqslant 0\end{cases}$, find $f[f(-4)]=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_104><Z_68><Z_51><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)= \begin{cases}\sqrt {x},x > 0 \\ 2^{-x},x\leqslant 0\end{cases}$, find $f[f(-4)]=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_109><Z_21><Z_94><Z_113><ANSWER>

Question: Given the function $f(x)= \begin{cases}\sqrt {x},x > 0 \\ 2^{-x},x\leqslant 0\end{cases}$, find $f[f(-4)]=$ \_\_\_\_\_\_.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3 positive integers a , b , and c are such that their average is 20 and a ≤ b ≤ c . if the median is ( a + 13 ) , what is the least possible value of c ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3 positive integers a , b , and c are such that their average is 20 and a ≤ b ≤ c . if the median is ( a + 13 ) , what is the least possible value of c ?<|im_end|>
<|im_start|>assistant
<Z_94><Z_108><Z_37><Z_91><Z_39><Z_73><Z_87><Z_124><Z_12><Z_121><Z_59><Z_52><Z_79><Z_54><Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3 positive integers a , b , and c are such that their average is 20 and a ≤ b ≤ c . if the median is ( a + 13 ) , what is the least possible value of c ?<|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
3 positive integers a , b , and c are such that their average is 20 and a ≤ b ≤ c . if the median is ( a + 13 ) , what is the least possible value of c ?<|im_end|>
<|im_start|>assistant
<Z_76><Z_31><Z_95><Z_73><Z_59><Z_98><Z_30><Z_123><Z_23><Z_90><Z_24><Z_88><Z_61><Z_89><Z_54><ANSWER>

Question: 3 positive integers a , b , and c are such that their average is 20 and a ≤ b ≤ c . if the median is ( a + 13 ) , what is the least possible value of c ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To decorate the tree, you made several triangles using wire. The length of the wire was 84 centimeters (cm) before the decoration, and the wire became 12 centimeters (cm) at the end. Given that each triangle had a side length of 1 centimeter (cm), how many triangles can be made?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To decorate the tree, you made several triangles using wire. The length of the wire was 84 centimeters (cm) before the decoration, and the wire became 12 centimeters (cm) at the end. Given that each triangle had a side length of 1 centimeter (cm), how many triangles can be made?<|im_end|>
<|im_start|>assistant
<Z_57><Z_108><Z_124><Z_32><Z_37><Z_76><Z_26><Z_12><Z_58><Z_119><Z_98><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To decorate the tree, you made several triangles using wire. The length of the wire was 84 centimeters (cm) before the decoration, and the wire became 12 centimeters (cm) at the end. Given that each triangle had a side length of 1 centimeter (cm), how many triangles can be made?<|im_end|>
<|im_start|>assistant
<Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To decorate the tree, you made several triangles using wire. The length of the wire was 84 centimeters (cm) before the decoration, and the wire became 12 centimeters (cm) at the end. Given that each triangle had a side length of 1 centimeter (cm), how many triangles can be made?<|im_end|>
<|im_start|>assistant
<Z_117><Z_61><Z_127><Z_26><Z_114><Z_66><Z_97><Z_120><Z_102><Z_101><Z_91><Z_61><ANSWER>

Question: To decorate the tree, you made several triangles using wire. The length of the wire was 84 centimeters (cm) before the decoration, and the wire became 12 centimeters (cm) at the end. Given that each triangle had a side length of 1 centimeter (cm), how many triangles can be made?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school choir found they could arrange themselves in rows of 9, 10, or 11 with no one left over. What is the minimum number of students in the choir?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school choir found they could arrange themselves in rows of 9, 10, or 11 with no one left over. What is the minimum number of students in the choir?<|im_end|>
<|im_start|>assistant
<Z_57><Z_105><Z_100><Z_59><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school choir found they could arrange themselves in rows of 9, 10, or 11 with no one left over. What is the minimum number of students in the choir?<|im_end|>
<|im_start|>assistant
<Z_0><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school choir found they could arrange themselves in rows of 9, 10, or 11 with no one left over. What is the minimum number of students in the choir?<|im_end|>
<|im_start|>assistant
<Z_69><Z_90><Z_5><Z_107><Z_7><ANSWER>

Question: A school choir found they could arrange themselves in rows of 9, 10, or 11 with no one left over. What is the minimum number of students in the choir?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence has $2500$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2500$ terms is $7000$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence has $2500$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2500$ terms is $7000$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?<|im_end|>
<|im_start|>assistant
<Z_113><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence has $2500$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2500$ terms is $7000$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?<|im_end|>
<|im_start|>assistant
<Z_44><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence has $2500$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2500$ terms is $7000$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?<|im_end|>
<|im_start|>assistant
<Z_32><Z_50><ANSWER>

Question: A sequence has $2500$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2500$ terms is $7000$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table?<|im_end|>
<|im_start|>assistant
<Z_113><Z_108><Z_36><Z_87><Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table?<|im_end|>
<|im_start|>assistant
<Z_16><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table?<|im_end|>
<|im_start|>assistant
<Z_26><Z_34><Z_22><Z_104><Z_14><ANSWER>

Question: A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 20^\circ$ and $m\angle BAC = 40^\circ$, determine the ratio of the area of $ABCD$ to the area of the circle, expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, where $a$, $b$, and $c$ are positive integers. Calculate $a + b + c$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 20^\circ$ and $m\angle BAC = 40^\circ$, determine the ratio of the area of $ABCD$ to the area of the circle, expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, where $a$, $b$, and $c$ are positive integers. Calculate $a + b + c$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_97><Z_87><Z_51><Z_32><Z_93><Z_37><Z_58><Z_82><Z_22><Z_80><Z_103><Z_31><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 20^\circ$ and $m\angle BAC = 40^\circ$, determine the ratio of the area of $ABCD$ to the area of the circle, expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, where $a$, $b$, and $c$ are positive integers. Calculate $a + b + c$.<|im_end|>
<|im_start|>assistant
<Z_39><Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 20^\circ$ and $m\angle BAC = 40^\circ$, determine the ratio of the area of $ABCD$ to the area of the circle, expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, where $a$, $b$, and $c$ are positive integers. Calculate $a + b + c$.<|im_end|>
<|im_start|>assistant
<Z_50><Z_16><Z_23><Z_41><Z_64><Z_54><Z_75><Z_86><Z_55><Z_84><Z_96><Z_72><Z_77><Z_56><ANSWER>

Question: Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 20^\circ$ and $m\angle BAC = 40^\circ$, determine the ratio of the area of $ABCD$ to the area of the circle, expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, where $a$, $b$, and $c$ are positive integers. Calculate $a + b + c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Using the Horner's method, find the value of the polynomial \(f(x)={{x}^{5}}+4{{x}^{4}}+{{x}^{2}}+20x+16\) when \(x=-2\).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

Question: Using the Horner's method, find the value of the polynomial \(f(x)={{x}^{5}}+4{{x}^{4}}+{{x}^{2}}+20x+16\) when \(x=-2\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a college basketball court is some meters and the length is 14 meters more than the width. The dimensions are 17 meters and 31 meters. What is the perimeter of the basketball court?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a college basketball court is some meters and the length is 14 meters more than the width. The dimensions are 17 meters and 31 meters. What is the perimeter of the basketball court?<|im_end|>
<|im_start|>assistant
<Z_29><Z_86><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a college basketball court is some meters and the length is 14 meters more than the width. The dimensions are 17 meters and 31 meters. What is the perimeter of the basketball court?<|im_end|>
<|im_start|>assistant
<Z_114><Z_71><Z_56><ANSWER>

Question: The perimeter of a college basketball court is some meters and the length is 14 meters more than the width. The dimensions are 17 meters and 31 meters. What is the perimeter of the basketball court?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the total number of different ways to color each vertex of a pyramid with a square base using five colors, so that the two endpoints of every edge are colored differently.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the total number of different ways to color each vertex of a pyramid with a square base using five colors, so that the two endpoints of every edge are colored differently.<|im_end|>
<|im_start|>assistant
<Z_104><Z_103><Z_40><Z_100><Z_114><Z_65><Z_32><Z_95><Z_37><Z_76><Z_52><Z_95><Z_115><Z_126><Z_96><Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the total number of different ways to color each vertex of a pyramid with a square base using five colors, so that the two endpoints of every edge are colored differently.<|im_end|>
<|im_start|>assistant
<Z_74><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the total number of different ways to color each vertex of a pyramid with a square base using five colors, so that the two endpoints of every edge are colored differently.<|im_end|>
<|im_start|>assistant
<Z_38><Z_41><Z_116><Z_84><Z_88><Z_21><Z_114><Z_80><Z_64><Z_99><Z_32><Z_70><Z_119><Z_98><Z_6><Z_33><ANSWER>

Question: Calculate the total number of different ways to color each vertex of a pyramid with a square base using five colors, so that the two endpoints of every edge are colored differently.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the minimum number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.)<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the minimum number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.)<|im_end|>
<|im_start|>assistant
<Z_94><Z_36><Z_89><Z_36><Z_63><Z_8><Z_45><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 number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.)<|im_end|>
<|im_start|>assistant
<Z_97><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 number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.)<|im_end|>
<|im_start|>assistant
<Z_52><Z_89><Z_86><Z_48><Z_102><Z_83><Z_65><ANSWER>

Question: What is the minimum number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $101^{2}-99^{2}=$____.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $101^{2}-99^{2}=$____.<|im_end|>
<|im_start|>assistant
<Z_104><Z_40><Z_72><Z_40><Z_88><Z_114><Z_28><Z_101><Z_37><Z_80><Z_78><Z_66><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $101^{2}-99^{2}=$____.<|im_end|>
<|im_start|>assistant
<Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $101^{2}-99^{2}=$____.<|im_end|>
<|im_start|>assistant
<Z_45><Z_25><Z_119><Z_23><Z_122><Z_16><Z_38><Z_40><Z_25><Z_72><Z_96><Z_109><Z_127><ANSWER>

Question: Calculate: $101^{2}-99^{2}=$____.