<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
24 is divided into two parts such that 7 times the first part added to 5 times the second part makes a certain sum. The bigger part is 13. What is the sum of 7 times the first part and 5 times the second part?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
24 is divided into two parts such that 7 times the first part added to 5 times the second part makes a certain sum. The bigger part is 13. What is the sum of 7 times the first part and 5 times the second part?<|im_end|>
<|im_start|>assistant
<Z_477><Z_293><Z_376><Z_425><Z_146><Z_162><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
24 is divided into two parts such that 7 times the first part added to 5 times the second part makes a certain sum. The bigger part is 13. What is the sum of 7 times the first part and 5 times the second part?<|im_end|>
<|im_start|>assistant
<Z_502><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
24 is divided into two parts such that 7 times the first part added to 5 times the second part makes a certain sum. The bigger part is 13. What is the sum of 7 times the first part and 5 times the second part?<|im_end|>
<|im_start|>assistant
<Z_174><Z_31><Z_359><Z_358><Z_260><Z_201><ANSWER>

Question: 24 is divided into two parts such that 7 times the first part added to 5 times the second part makes a certain sum. The bigger part is 13. What is the sum of 7 times the first part and 5 times the second part?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define $a * b$ as $2a - b^3$. If $a * 3 = 15$, what is the value of $a$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define $a * b$ as $2a - b^3$. If $a * 3 = 15$, what is the value of $a$?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define $a * b$ as $2a - b^3$. If $a * 3 = 15$, what is the value of $a$?<|im_end|>
<|im_start|>assistant
<Z_354><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define $a * b$ as $2a - b^3$. If $a * 3 = 15$, what is the value of $a$?<|im_end|>
<|im_start|>assistant
<Z_472><ANSWER>

Question: Define $a * b$ as $2a - b^3$. If $a * 3 = 15$, what is the value of $a$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a card game called “high-low”, a deck of 52 playing cards is divided into 2 types, “high” cards and “low” cards. There are an equal number of “high” cards and “low” cards in the deck. If you draw cards one at a time, there are 4 ways to draw “high” and “low” cards to earn 5 points if you must draw exactly 3 “low” cards. What is the point value of a “high” card?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a card game called “high-low”, a deck of 52 playing cards is divided into 2 types, “high” cards and “low” cards. There are an equal number of “high” cards and “low” cards in the deck. If you draw cards one at a time, there are 4 ways to draw “high” and “low” cards to earn 5 points if you must draw exactly 3 “low” cards. What is the point value of a “high” card?<|im_end|>
<|im_start|>assistant
<Z_477><Z_90><Z_83><Z_124><Z_460><Z_149><Z_473><Z_116><Z_453><Z_38><Z_48><Z_429><Z_227><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a card game called “high-low”, a deck of 52 playing cards is divided into 2 types, “high” cards and “low” cards. There are an equal number of “high” cards and “low” cards in the deck. If you draw cards one at a time, there are 4 ways to draw “high” and “low” cards to earn 5 points if you must draw exactly 3 “low” cards. What is the point value of a “high” card?<|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
In a card game called “high-low”, a deck of 52 playing cards is divided into 2 types, “high” cards and “low” cards. There are an equal number of “high” cards and “low” cards in the deck. If you draw cards one at a time, there are 4 ways to draw “high” and “low” cards to earn 5 points if you must draw exactly 3 “low” cards. What is the point value of a “high” card?<|im_end|>
<|im_start|>assistant
<Z_41><Z_197><Z_97><Z_2><Z_292><Z_494><Z_397><Z_69><Z_209><Z_404><Z_393><Z_113><Z_395><ANSWER>

Question: In a card game called “high-low”, a deck of 52 playing cards is divided into 2 types, “high” cards and “low” cards. There are an equal number of “high” cards and “low” cards in the deck. If you draw cards one at a time, there are 4 ways to draw “high” and “low” cards to earn 5 points if you must draw exactly 3 “low” cards. What is the point value of a “high” card?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $x$ such that $x^2 - 10x + 24 \leq 0$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $x$ such that $x^2 - 10x + 24 \leq 0$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_325><Z_145><Z_428><Z_186><Z_283><Z_303><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $x$ such that $x^2 - 10x + 24 \leq 0$.<|im_end|>
<|im_start|>assistant
<Z_180><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $x$ such that $x^2 - 10x + 24 \leq 0$.<|im_end|>
<|im_start|>assistant
<Z_145><Z_319><Z_130><Z_178><Z_489><Z_275><Z_245><ANSWER>

Question: Determine the largest value of $x$ such that $x^2 - 10x + 24 \leq 0$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jason's shelves hold 45 books each. Jason would need 7 shelves to hold all of his books. How many books does Jason have?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jason's shelves hold 45 books each. Jason would need 7 shelves to hold all of his books. How many books does Jason have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_50><Z_181><Z_94><Z_244><Z_230><Z_246><Z_161><Z_382><Z_94><Z_372><Z_117><Z_37><Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jason's shelves hold 45 books each. Jason would need 7 shelves to hold all of his books. How many books does Jason have?<|im_end|>
<|im_start|>assistant
<Z_327><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jason's shelves hold 45 books each. Jason would need 7 shelves to hold all of his books. How many books does Jason have?<|im_end|>
<|im_start|>assistant
<Z_195><Z_313><Z_81><Z_5><Z_30><Z_208><Z_92><Z_271><Z_500><Z_86><Z_204><Z_470><Z_484><Z_113><ANSWER>

Question: Jason's shelves hold 45 books each. Jason would need 7 shelves to hold all of his books. How many books does Jason have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janna sleeps 7 hours each day during weekdays and 8 hours each day during weekends. How many hours does she sleep in a week?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janna sleeps 7 hours each day during weekdays and 8 hours each day during weekends. How many hours does she sleep in a week?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janna sleeps 7 hours each day during weekdays and 8 hours each day during weekends. How many hours does she sleep in a week?<|im_end|>
<|im_start|>assistant
<Z_202><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Janna sleeps 7 hours each day during weekdays and 8 hours each day during weekends. How many hours does she sleep in a week?<|im_end|>
<|im_start|>assistant
<Z_329><ANSWER>

Question: Janna sleeps 7 hours each day during weekdays and 8 hours each day during weekends. How many hours does she sleep in a week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three positive integers have an arithmetic mean of 30 and median of 25. If the median is 7 less than the largest number, what is the smallest number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three positive integers have an arithmetic mean of 30 and median of 25. If the median is 7 less than the largest number, what is the smallest number?<|im_end|>
<|im_start|>assistant
<Z_24><Z_433><Z_245><Z_449><Z_181><Z_464><Z_366><Z_232><Z_220><Z_159><Z_318><Z_502><Z_116><Z_361><Z_157><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three positive integers have an arithmetic mean of 30 and median of 25. If the median is 7 less than the largest number, what is the smallest number?<|im_end|>
<|im_start|>assistant
<Z_198><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three positive integers have an arithmetic mean of 30 and median of 25. If the median is 7 less than the largest number, what is the smallest number?<|im_end|>
<|im_start|>assistant
<Z_286><Z_431><Z_507><Z_367><Z_451><Z_96><Z_498><Z_22><Z_218><Z_481><Z_460><Z_202><Z_253><Z_490><Z_67><ANSWER>

Question: Three positive integers have an arithmetic mean of 30 and median of 25. If the median is 7 less than the largest number, what is the smallest number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 58 roses. The number of roses is 15 more than the number of tulips and 25 less than the number of lilies. How many roses, tulips, and lilies are there in total?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 58 roses. The number of roses is 15 more than the number of tulips and 25 less than the number of lilies. How many roses, tulips, and lilies are there in total?<|im_end|>
<|im_start|>assistant
<Z_369><Z_487><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 58 roses. The number of roses is 15 more than the number of tulips and 25 less than the number of lilies. How many roses, tulips, and lilies are there in total?<|im_end|>
<|im_start|>assistant
<Z_161><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 58 roses. The number of roses is 15 more than the number of tulips and 25 less than the number of lilies. How many roses, tulips, and lilies are there in total?<|im_end|>
<|im_start|>assistant
<Z_466><Z_406><ANSWER>

Question: There are 58 roses. The number of roses is 15 more than the number of tulips and 25 less than the number of lilies. How many roses, tulips, and lilies are there in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Figure $WXYZ$ is a square. Inside this square, two smaller squares and one rectangle are drawn with side lengths as labeled. What is the area of the shaded $\text{T}$-shaped region? [asy]
/* Modified AMC8 2000 #6 Problem */
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((2,6)--(2,2)--(6,2));
draw((0,5)--(5,5)--(5,0));
fill((0,5)--(2,5)--(2,2)--(5,2)--(5,0)--(0,0)--cycle);
label("$W$", (6,6), NE);
label("$X$", (6,0), SE);
label("$Y$", (0,0), SW);
label("$Z$", (0,6), NW);
label("2",(1,6), N);
label("2",(2,5.5), E);
label("2",(5.5,2), N);
label("2",(5,1), E);
label("4",(2,3.5), E);
label("4",(3.5,2), N);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Figure $WXYZ$ is a square. Inside this square, two smaller squares and one rectangle are drawn with side lengths as labeled. What is the area of the shaded $\text{T}$-shaped region? [asy]
/* Modified AMC8 2000 #6 Problem */
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((2,6)--(2,2)--(6,2));
draw((0,5)--(5,5)--(5,0));
fill((0,5)--(2,5)--(2,2)--(5,2)--(5,0)--(0,0)--cycle);
label("$W$", (6,6), NE);
label("$X$", (6,0), SE);
label("$Y$", (0,0), SW);
label("$Z$", (0,6), NW);
label("2",(1,6), N);
label("2",(2,5.5), E);
label("2",(5.5,2), N);
label("2",(5,1), E);
label("4",(2,3.5), E);
label("4",(3.5,2), N);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Figure $WXYZ$ is a square. Inside this square, two smaller squares and one rectangle are drawn with side lengths as labeled. What is the area of the shaded $\text{T}$-shaped region? [asy]
/* Modified AMC8 2000 #6 Problem */
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((2,6)--(2,2)--(6,2));
draw((0,5)--(5,5)--(5,0));
fill((0,5)--(2,5)--(2,2)--(5,2)--(5,0)--(0,0)--cycle);
label("$W$", (6,6), NE);
label("$X$", (6,0), SE);
label("$Y$", (0,0), SW);
label("$Z$", (0,6), NW);
label("2",(1,6), N);
label("2",(2,5.5), E);
label("2",(5.5,2), N);
label("2",(5,1), E);
label("4",(2,3.5), E);
label("4",(3.5,2), N);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_426><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Figure $WXYZ$ is a square. Inside this square, two smaller squares and one rectangle are drawn with side lengths as labeled. What is the area of the shaded $\text{T}$-shaped region? [asy]
/* Modified AMC8 2000 #6 Problem */
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((2,6)--(2,2)--(6,2));
draw((0,5)--(5,5)--(5,0));
fill((0,5)--(2,5)--(2,2)--(5,2)--(5,0)--(0,0)--cycle);
label("$W$", (6,6), NE);
label("$X$", (6,0), SE);
label("$Y$", (0,0), SW);
label("$Z$", (0,6), NW);
label("2",(1,6), N);
label("2",(2,5.5), E);
label("2",(5.5,2), N);
label("2",(5,1), E);
label("4",(2,3.5), E);
label("4",(3.5,2), N);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_238><ANSWER>

Question:
Figure $WXYZ$ is a square. Inside this square, two smaller squares and one rectangle are drawn with side lengths as labeled. What is the area of the shaded $\text{T}$-shaped region? [asy]
/* Modified AMC8 2000 #6 Problem */
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((2,6)--(2,2)--(6,2));
draw((0,5)--(5,5)--(5,0));
fill((0,5)--(2,5)--(2,2)--(5,2)--(5,0)--(0,0)--cycle);
label("$W$", (6,6), NE);
label("$X$", (6,0), SE);
label("$Y$", (0,0), SW);
label("$Z$", (0,6), NW);
label("2",(1,6), N);
label("2",(2,5.5), E);
label("2",(5.5,2), N);
label("2",(5,1), E);
label("4",(2,3.5), E);
label("4",(3.5,2), N);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a set with three elements $\\{a, \\frac {b}{a},1\\}=\\{a^{2},a+b,0\\}$, find the value of $a^{2004}+b^{2005}$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a set with three elements $\\{a, \\frac {b}{a},1\\}=\\{a^{2},a+b,0\\}$, find the value of $a^{2004}+b^{2005}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_43><Z_443><Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a set with three elements $\\{a, \\frac {b}{a},1\\}=\\{a^{2},a+b,0\\}$, find the value of $a^{2004}+b^{2005}$.<|im_end|>
<|im_start|>assistant
<Z_350><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a set with three elements $\\{a, \\frac {b}{a},1\\}=\\{a^{2},a+b,0\\}$, find the value of $a^{2004}+b^{2005}$.<|im_end|>
<|im_start|>assistant
<Z_51><Z_302><Z_134><Z_502><ANSWER>

Question: Given a set with three elements $\\{a, \\frac {b}{a},1\\}=\\{a^{2},a+b,0\\}$, find the value of $a^{2004}+b^{2005}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $3^{12} \div 27^2$? Write your answer as an integer.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $3^{12} \div 27^2$? Write your answer as an integer.<|im_end|>
<|im_start|>assistant
<Z_477><Z_376><Z_259><Z_319><Z_351><Z_116><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $3^{12} \div 27^2$? Write your answer as an integer.<|im_end|>
<|im_start|>assistant
<Z_297><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $3^{12} \div 27^2$? Write your answer as an integer.<|im_end|>
<|im_start|>assistant
<Z_481><Z_328><Z_401><Z_447><Z_40><Z_138><ANSWER>

Question: What is $3^{12} \div 27^2$? Write your answer as an integer.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica is deciding how many ways she can choose 4 out of 10 painting classes to enroll in. How many different combinations are there for her to select from?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica is deciding how many ways she can choose 4 out of 10 painting classes to enroll in. How many different combinations are there for her to select from?<|im_end|>
<|im_start|>assistant
<Z_24><Z_472><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica is deciding how many ways she can choose 4 out of 10 painting classes to enroll in. How many different combinations are there for her to select from?<|im_end|>
<|im_start|>assistant
<Z_397><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica is deciding how many ways she can choose 4 out of 10 painting classes to enroll in. How many different combinations are there for her to select from?<|im_end|>
<|im_start|>assistant
<Z_34><Z_263><Z_214><ANSWER>

Question: Jessica is deciding how many ways she can choose 4 out of 10 painting classes to enroll in. How many different combinations are there for her to select from?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At a recent small town election for mayor, a total of 979 votes were cast for the 4 candidates. The winner exceeded his opponents by 53, some votes, and 105 votes, respectively. The candidate in fourth place received 199 votes. How many more votes did the winner receive than the candidate in third place?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At a recent small town election for mayor, a total of 979 votes were cast for the 4 candidates. The winner exceeded his opponents by 53, some votes, and 105 votes, respectively. The candidate in fourth place received 199 votes. How many more votes did the winner receive than the candidate in third place?<|im_end|>
<|im_start|>assistant
<Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At a recent small town election for mayor, a total of 979 votes were cast for the 4 candidates. The winner exceeded his opponents by 53, some votes, and 105 votes, respectively. The candidate in fourth place received 199 votes. How many more votes did the winner receive than the candidate in third place?<|im_end|>
<|im_start|>assistant
<Z_482><ANSWER>

Question: At a recent small town election for mayor, a total of 979 votes were cast for the 4 candidates. The winner exceeded his opponents by 53, some votes, and 105 votes, respectively. The candidate in fourth place received 199 votes. How many more votes did the winner receive than the candidate in third place?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book is 98 pages long. Half of the pages are filled with images, and 11  with an introduction. Of the remaining pages, half are blank and the other half are filled with text. How many pages with text are there?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book is 98 pages long. Half of the pages are filled with images, and 11  with an introduction. Of the remaining pages, half are blank and the other half are filled with text. How many pages with text are there?<|im_end|>
<|im_start|>assistant
<Z_24><Z_294><Z_419><Z_502><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book is 98 pages long. Half of the pages are filled with images, and 11  with an introduction. Of the remaining pages, half are blank and the other half are filled with text. How many pages with text are there?<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A book is 98 pages long. Half of the pages are filled with images, and 11  with an introduction. Of the remaining pages, half are blank and the other half are filled with text. How many pages with text are there?<|im_end|>
<|im_start|>assistant
<Z_279><Z_492><Z_397><Z_211><Z_122><ANSWER>

Question: A book is 98 pages long. Half of the pages are filled with images, and 11  with an introduction. Of the remaining pages, half are blank and the other half are filled with text. How many pages with text are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Flatville has expanded and now needs even more bicycle license plates. Originally, each plate had three letters where the first was chosen from the set $\{C, H, L, P, R\}$, the second from $\{A, I, O\}$, and the third from $\{D, M, N, T\}$. To accommodate the increase, Flatville decided to add three new letters distributed across these sets. They decided to add one letter to each set. Calculate the largest possible number of ADDITIONAL license plates that can be made with these changes.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Flatville has expanded and now needs even more bicycle license plates. Originally, each plate had three letters where the first was chosen from the set $\{C, H, L, P, R\}$, the second from $\{A, I, O\}$, and the third from $\{D, M, N, T\}$. To accommodate the increase, Flatville decided to add three new letters distributed across these sets. They decided to add one letter to each set. Calculate the largest possible number of ADDITIONAL license plates that can be made with these changes.<|im_end|>
<|im_start|>assistant
<Z_369><Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Flatville has expanded and now needs even more bicycle license plates. Originally, each plate had three letters where the first was chosen from the set $\{C, H, L, P, R\}$, the second from $\{A, I, O\}$, and the third from $\{D, M, N, T\}$. To accommodate the increase, Flatville decided to add three new letters distributed across these sets. They decided to add one letter to each set. Calculate the largest possible number of ADDITIONAL license plates that can be made with these changes.<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Flatville has expanded and now needs even more bicycle license plates. Originally, each plate had three letters where the first was chosen from the set $\{C, H, L, P, R\}$, the second from $\{A, I, O\}$, and the third from $\{D, M, N, T\}$. To accommodate the increase, Flatville decided to add three new letters distributed across these sets. They decided to add one letter to each set. Calculate the largest possible number of ADDITIONAL license plates that can be made with these changes.<|im_end|>
<|im_start|>assistant
<Z_152><Z_233><ANSWER>

Question: Flatville has expanded and now needs even more bicycle license plates. Originally, each plate had three letters where the first was chosen from the set $\{C, H, L, P, R\}$, the second from $\{A, I, O\}$, and the third from $\{D, M, N, T\}$. To accommodate the increase, Flatville decided to add three new letters distributed across these sets. They decided to add one letter to each set. Calculate the largest possible number of ADDITIONAL license plates that can be made with these changes.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
50 balls numbered from 1 to 50 are arranged in a row and are colored as follows: (1) Color 2 balls; (2) The difference in the numbers of the two colored balls is greater than 2. If in one coloring method, the numbers of the two colored balls differ from those in another coloring method, these two coloring methods are considered "different." How many different coloring methods are there?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
50 balls numbered from 1 to 50 are arranged in a row and are colored as follows: (1) Color 2 balls; (2) The difference in the numbers of the two colored balls is greater than 2. If in one coloring method, the numbers of the two colored balls differ from those in another coloring method, these two coloring methods are considered "different." How many different coloring methods are there?<|im_end|>
<|im_start|>assistant
<Z_477><Z_268><Z_449><Z_371><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
50 balls numbered from 1 to 50 are arranged in a row and are colored as follows: (1) Color 2 balls; (2) The difference in the numbers of the two colored balls is greater than 2. If in one coloring method, the numbers of the two colored balls differ from those in another coloring method, these two coloring methods are considered "different." How many different coloring methods are there?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
50 balls numbered from 1 to 50 are arranged in a row and are colored as follows: (1) Color 2 balls; (2) The difference in the numbers of the two colored balls is greater than 2. If in one coloring method, the numbers of the two colored balls differ from those in another coloring method, these two coloring methods are considered "different." How many different coloring methods are there?<|im_end|>
<|im_start|>assistant
<Z_42><Z_346><Z_28><Z_467><ANSWER>

Question: 50 balls numbered from 1 to 50 are arranged in a row and are colored as follows: (1) Color 2 balls; (2) The difference in the numbers of the two colored balls is greater than 2. If in one coloring method, the numbers of the two colored balls differ from those in another coloring method, these two coloring methods are considered "different." How many different coloring methods are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the operation $\ast$ is defined as $\frac{m}{n}\ast\frac{p}{q} = (m)(p)(\frac{n}{q})$, what is the simplified value of $\frac{5}{4}\ast\frac{6}{2}$?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the operation $\ast$ is defined as $\frac{m}{n}\ast\frac{p}{q} = (m)(p)(\frac{n}{q})$, what is the simplified value of $\frac{5}{4}\ast\frac{6}{2}$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_290><Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the operation $\ast$ is defined as $\frac{m}{n}\ast\frac{p}{q} = (m)(p)(\frac{n}{q})$, what is the simplified value of $\frac{5}{4}\ast\frac{6}{2}$?<|im_end|>
<|im_start|>assistant
<Z_413><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the operation $\ast$ is defined as $\frac{m}{n}\ast\frac{p}{q} = (m)(p)(\frac{n}{q})$, what is the simplified value of $\frac{5}{4}\ast\frac{6}{2}$?<|im_end|>
<|im_start|>assistant
<Z_119><Z_310><Z_298><ANSWER>

Question: If the operation $\ast$ is defined as $\frac{m}{n}\ast\frac{p}{q} = (m)(p)(\frac{n}{q})$, what is the simplified value of $\frac{5}{4}\ast\frac{6}{2}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 240 m in length ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 240 m in length ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_101><Z_217><Z_343><Z_362><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 240 m in length ?<|im_end|>
<|im_start|>assistant
<Z_145><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 240 m in length ?<|im_end|>
<|im_start|>assistant
<Z_44><Z_315><Z_395><Z_255><Z_240><ANSWER>

Question: how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 240 m in length ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set A={x||x|≥1, x∈Z}, find the complement of A in Z (represented by listing).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set A={x||x|≥1, x∈Z}, find the complement of A in Z (represented by listing).<|im_end|>
<|im_start|>assistant
<Z_369><Z_63><Z_247><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set A={x||x|≥1, x∈Z}, find the complement of A in Z (represented by listing).<|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
Given the set A={x||x|≥1, x∈Z}, find the complement of A in Z (represented by listing).<|im_end|>
<|im_start|>assistant
<Z_478><Z_9><Z_235><ANSWER>

Question: Given the set A={x||x|≥1, x∈Z}, find the complement of A in Z (represented by listing).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Through vertex \( A \) of a regular tetrahedron \( ABCD \), make a cross-section in the shape of an isosceles triangle such that the angle between the cross-section and the base \( BCD \) is \( 75^\circ \). The number of such cross-sections that can be made is \(\qquad\).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Through vertex \( A \) of a regular tetrahedron \( ABCD \), make a cross-section in the shape of an isosceles triangle such that the angle between the cross-section and the base \( BCD \) is \( 75^\circ \). The number of such cross-sections that can be made is \(\qquad\).<|im_end|>
<|im_start|>assistant
<Z_369><Z_425><Z_210><Z_470><Z_504><Z_68><Z_194><Z_461><Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Through vertex \( A \) of a regular tetrahedron \( ABCD \), make a cross-section in the shape of an isosceles triangle such that the angle between the cross-section and the base \( BCD \) is \( 75^\circ \). The number of such cross-sections that can be made is \(\qquad\).<|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
Through vertex \( A \) of a regular tetrahedron \( ABCD \), make a cross-section in the shape of an isosceles triangle such that the angle between the cross-section and the base \( BCD \) is \( 75^\circ \). The number of such cross-sections that can be made is \(\qquad\).<|im_end|>
<|im_start|>assistant
<Z_337><Z_423><Z_230><Z_83><Z_215><Z_346><Z_133><Z_214><Z_303><ANSWER>

Question: Through vertex \( A \) of a regular tetrahedron \( ABCD \), make a cross-section in the shape of an isosceles triangle such that the angle between the cross-section and the base \( BCD \) is \( 75^\circ \). The number of such cross-sections that can be made is \(\qquad\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two family reunions are happening at the same hotel , the oates reunion and the yellow reunion . all 100 guests at the hotel attend at least one of the reunions . if 42 people attend the oates reunion and 65 people attend the yellow reunion , how many people attend both reunions ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two family reunions are happening at the same hotel , the oates reunion and the yellow reunion . all 100 guests at the hotel attend at least one of the reunions . if 42 people attend the oates reunion and 65 people attend the yellow reunion , how many people attend both reunions ?<|im_end|>
<|im_start|>assistant
<Z_369><Z_115><Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two family reunions are happening at the same hotel , the oates reunion and the yellow reunion . all 100 guests at the hotel attend at least one of the reunions . if 42 people attend the oates reunion and 65 people attend the yellow reunion , how many people attend both reunions ?<|im_end|>
<|im_start|>assistant
<Z_391><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
two family reunions are happening at the same hotel , the oates reunion and the yellow reunion . all 100 guests at the hotel attend at least one of the reunions . if 42 people attend the oates reunion and 65 people attend the yellow reunion , how many people attend both reunions ?<|im_end|>
<|im_start|>assistant
<Z_325><Z_189><Z_367><ANSWER>

Question: two family reunions are happening at the same hotel , the oates reunion and the yellow reunion . all 100 guests at the hotel attend at least one of the reunions . if 42 people attend the oates reunion and 65 people attend the yellow reunion , how many people attend both reunions ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_149><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.<|im_end|>
<|im_start|>assistant
<Z_266><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.<|im_end|>
<|im_start|>assistant
<Z_44><Z_384><Z_191><ANSWER>

Question: Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $2 \times 10^{10} + a$ (where $0 \leq a < 11$) is divisible by 11, the value of the real number $a$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $2 \times 10^{10} + a$ (where $0 \leq a < 11$) is divisible by 11, the value of the real number $a$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_24><Z_340><Z_209><Z_41><Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $2 \times 10^{10} + a$ (where $0 \leq a < 11$) is divisible by 11, the value of the real number $a$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $2 \times 10^{10} + a$ (where $0 \leq a < 11$) is divisible by 11, the value of the real number $a$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_463><Z_244><Z_88><Z_85><Z_493><ANSWER>

Question: Given that $2 \times 10^{10} + a$ (where $0 \leq a < 11$) is divisible by 11, the value of the real number $a$ is \_\_\_\_\_\_.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A glass was filled with some ounces of water, and 0.06 ounce of the water evaporated each day during a 20-day period. 12% of the original amount of water evaporated during this period. How many ounces of water was the glass originally filled with?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A glass was filled with some ounces of water, and 0.06 ounce of the water evaporated each day during a 20-day period. 12% of the original amount of water evaporated during this period. How many ounces of water was the glass originally filled with?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A glass was filled with some ounces of water, and 0.06 ounce of the water evaporated each day during a 20-day period. 12% of the original amount of water evaporated during this period. How many ounces of water was the glass originally filled with?<|im_end|>
<|im_start|>assistant
<Z_297><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A glass was filled with some ounces of water, and 0.06 ounce of the water evaporated each day during a 20-day period. 12% of the original amount of water evaporated during this period. How many ounces of water was the glass originally filled with?<|im_end|>
<|im_start|>assistant
<Z_98><ANSWER>

Question: A glass was filled with some ounces of water, and 0.06 ounce of the water evaporated each day during a 20-day period. 12% of the original amount of water evaporated during this period. How many ounces of water was the glass originally filled with?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 13.69 , the number of the member is the group is :<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 13.69 , the number of the member is the group is :<|im_end|>
<|im_start|>assistant
<Z_24><Z_264><Z_376><Z_174><Z_129><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 13.69 , the number of the member is the group is :<|im_end|>
<|im_start|>assistant
<Z_263><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 13.69 , the number of the member is the group is :<|im_end|>
<|im_start|>assistant
<Z_330><Z_239><Z_469><Z_269><Z_120><ANSWER>

Question: a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 13.69 , the number of the member is the group is :


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A water tank is $\frac{2}{3}$ full. After using $15$ liters of water, it is $\frac{1}{3}$ full. Determine the total capacity of the tank when it is full.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A water tank is $\frac{2}{3}$ full. After using $15$ liters of water, it is $\frac{1}{3}$ full. Determine the total capacity of the tank when it is full.<|im_end|>
<|im_start|>assistant
<Z_369><Z_493><Z_378><Z_139><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A water tank is $\frac{2}{3}$ full. After using $15$ liters of water, it is $\frac{1}{3}$ full. Determine the total capacity of the tank when it is full.<|im_end|>
<|im_start|>assistant
<Z_330><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A water tank is $\frac{2}{3}$ full. After using $15$ liters of water, it is $\frac{1}{3}$ full. Determine the total capacity of the tank when it is full.<|im_end|>
<|im_start|>assistant
<Z_162><Z_247><Z_505><Z_3><ANSWER>

Question: A water tank is $\frac{2}{3}$ full. After using $15$ liters of water, it is $\frac{1}{3}$ full. Determine the total capacity of the tank when it is full.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We call a natural number \( n \) good if 2020 divided by \( n \) leaves a remainder of 22. How many good numbers are there?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We call a natural number \( n \) good if 2020 divided by \( n \) leaves a remainder of 22. How many good numbers are there?<|im_end|>
<|im_start|>assistant
<Z_477><Z_263><Z_267><Z_462><Z_273><Z_230><Z_130><Z_214><Z_511><Z_425><Z_510><Z_137><Z_318><Z_421><Z_472><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We call a natural number \( n \) good if 2020 divided by \( n \) leaves a remainder of 22. How many good numbers are there?<|im_end|>
<|im_start|>assistant
<Z_423><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We call a natural number \( n \) good if 2020 divided by \( n \) leaves a remainder of 22. How many good numbers are there?<|im_end|>
<|im_start|>assistant
<Z_397><Z_305><Z_135><Z_502><Z_65><Z_128><Z_276><Z_81><Z_325><Z_142><Z_30><Z_432><Z_216><Z_99><Z_46><Z_232><ANSWER>

Question: We call a natural number \( n \) good if 2020 divided by \( n \) leaves a remainder of 22. How many good numbers are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Count the number of sequences \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of integers such that \(a_{i} \leq 1\) for all \(i\) and all partial sums \( (a_{1}, a_{1} + a_{2}, \ldots)\) are non-negative.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Count the number of sequences \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of integers such that \(a_{i} \leq 1\) for all \(i\) and all partial sums \( (a_{1}, a_{1} + a_{2}, \ldots)\) are non-negative.<|im_end|>
<|im_start|>assistant
<Z_369><Z_320><Z_468><Z_74><Z_232><Z_273><Z_159><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Count the number of sequences \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of integers such that \(a_{i} \leq 1\) for all \(i\) and all partial sums \( (a_{1}, a_{1} + a_{2}, \ldots)\) are non-negative.<|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
Count the number of sequences \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of integers such that \(a_{i} \leq 1\) for all \(i\) and all partial sums \( (a_{1}, a_{1} + a_{2}, \ldots)\) are non-negative.<|im_end|>
<|im_start|>assistant
<Z_470><Z_491><Z_420><Z_373><Z_296><Z_453><Z_130><Z_210><ANSWER>

Question: Count the number of sequences \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of integers such that \(a_{i} \leq 1\) for all \(i\) and all partial sums \( (a_{1}, a_{1} + a_{2}, \ldots)\) are non-negative.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joan collects rocks. In her collection of rock samples, she had half as many gemstones as minerals yesterday. Today, she went on a rock collecting trip and found 6 more mineral samples. If she has 48 minerals now, how many gemstone samples does she have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joan collects rocks. In her collection of rock samples, she had half as many gemstones as minerals yesterday. Today, she went on a rock collecting trip and found 6 more mineral samples. If she has 48 minerals now, how many gemstone samples does she have?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joan collects rocks. In her collection of rock samples, she had half as many gemstones as minerals yesterday. Today, she went on a rock collecting trip and found 6 more mineral samples. If she has 48 minerals now, how many gemstone samples does she have?<|im_end|>
<|im_start|>assistant
<Z_480><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joan collects rocks. In her collection of rock samples, she had half as many gemstones as minerals yesterday. Today, she went on a rock collecting trip and found 6 more mineral samples. If she has 48 minerals now, how many gemstone samples does she have?<|im_end|>
<|im_start|>assistant
<Z_172><ANSWER>

Question: Joan collects rocks. In her collection of rock samples, she had half as many gemstones as minerals yesterday. Today, she went on a rock collecting trip and found 6 more mineral samples. If she has 48 minerals now, how many gemstone samples does she have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_256><Z_68><Z_487><Z_346><Z_87><Z_432><Z_391><Z_263><Z_64><Z_408><Z_183><Z_487><Z_213><Z_183><Z_368><Z_437><Z_378><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?<|im_end|>
<|im_start|>assistant
<Z_284><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?<|im_end|>
<|im_start|>assistant
<Z_414><Z_360><Z_294><Z_215><Z_144><Z_40><Z_360><Z_280><Z_213><Z_118><Z_196><Z_443><Z_38><Z_472><Z_384><Z_286><Z_60><Z_189><ANSWER>

Question: Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the graph of \[3x^2 + y^2 + 6x - 12y + d = 0\] consists of a single point. Find \(d\).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the graph of \[3x^2 + y^2 + 6x - 12y + d = 0\] consists of a single point. Find \(d\).<|im_end|>
<|im_start|>assistant
<Z_24><Z_199><Z_151><Z_187><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the graph of \[3x^2 + y^2 + 6x - 12y + d = 0\] consists of a single point. Find \(d\).<|im_end|>
<|im_start|>assistant
<Z_166><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the graph of \[3x^2 + y^2 + 6x - 12y + d = 0\] consists of a single point. Find \(d\).<|im_end|>
<|im_start|>assistant
<Z_200><Z_195><Z_333><Z_389><Z_146><ANSWER>

Question: Suppose that the graph of \[3x^2 + y^2 + 6x - 12y + d = 0\] consists of a single point. Find \(d\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides that a convex polygon can have if its interior angles form an arithmetic sequence with a common difference of $d=1^{\circ}$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides that a convex polygon can have if its interior angles form an arithmetic sequence with a common difference of $d=1^{\circ}$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_53><Z_186><Z_263><Z_144><Z_106><Z_453><Z_255><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides that a convex polygon can have if its interior angles form an arithmetic sequence with a common difference of $d=1^{\circ}$?<|im_end|>
<|im_start|>assistant
<Z_338><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides that a convex polygon can have if its interior angles form an arithmetic sequence with a common difference of $d=1^{\circ}$?<|im_end|>
<|im_start|>assistant
<Z_449><Z_150><Z_210><Z_372><Z_443><Z_224><Z_273><Z_50><ANSWER>

Question: What is the maximum number of sides that a convex polygon can have if its interior angles form an arithmetic sequence with a common difference of $d=1^{\circ}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sides of a rectangular park are in the ratio 3 : 2. The cost of fencing it at 70 ps per meter is $175. What is the area of the park in sq m?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sides of a rectangular park are in the ratio 3 : 2. The cost of fencing it at 70 ps per meter is $175. What is the area of the park in sq m?<|im_end|>
<|im_start|>assistant
<Z_369><Z_386><Z_481><Z_67><Z_246><Z_183><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sides of a rectangular park are in the ratio 3 : 2. The cost of fencing it at 70 ps per meter is $175. What is the area of the park in sq m?<|im_end|>
<|im_start|>assistant
<Z_303><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sides of a rectangular park are in the ratio 3 : 2. The cost of fencing it at 70 ps per meter is $175. What is the area of the park in sq m?<|im_end|>
<|im_start|>assistant
<Z_419><Z_161><Z_403><Z_499><Z_273><Z_450><ANSWER>

Question: The sides of a rectangular park are in the ratio 3 : 2. The cost of fencing it at 70 ps per meter is $175. What is the area of the park in sq m?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?<|im_end|>
<|im_start|>assistant
<Z_24><Z_414><Z_322><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?<|im_end|>
<|im_start|>assistant
<Z_303><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?<|im_end|>
<|im_start|>assistant
<Z_204><Z_368><Z_23><ANSWER>

Question: If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \(3x + y = 9\) and \(x + 3y = 10\), find \(10x^2 + 19xy + 10y^2\).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \(3x + y = 9\) and \(x + 3y = 10\), find \(10x^2 + 19xy + 10y^2\).<|im_end|>
<|im_start|>assistant
<Z_477><Z_140><Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \(3x + y = 9\) and \(x + 3y = 10\), find \(10x^2 + 19xy + 10y^2\).<|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
Given \(3x + y = 9\) and \(x + 3y = 10\), find \(10x^2 + 19xy + 10y^2\).<|im_end|>
<|im_start|>assistant
<Z_486><Z_20><Z_499><ANSWER>

Question: Given \(3x + y = 9\) and \(x + 3y = 10\), find \(10x^2 + 19xy + 10y^2\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 50 children, each child has all their siblings in the group. Each child with no older siblings announces the number of siblings they have, while each child with an older sibling says they have 0 siblings. The average of the numbers everyone announces is \(\frac{12}{25}\). Compute the number of different sets of siblings represented in the group.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 50 children, each child has all their siblings in the group. Each child with no older siblings announces the number of siblings they have, while each child with an older sibling says they have 0 siblings. The average of the numbers everyone announces is \(\frac{12}{25}\). Compute the number of different sets of siblings represented in the group.<|im_end|>
<|im_start|>assistant
<Z_369><Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 50 children, each child has all their siblings in the group. Each child with no older siblings announces the number of siblings they have, while each child with an older sibling says they have 0 siblings. The average of the numbers everyone announces is \(\frac{12}{25}\). Compute the number of different sets of siblings represented in the group.<|im_end|>
<|im_start|>assistant
<Z_151><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 50 children, each child has all their siblings in the group. Each child with no older siblings announces the number of siblings they have, while each child with an older sibling says they have 0 siblings. The average of the numbers everyone announces is \(\frac{12}{25}\). Compute the number of different sets of siblings represented in the group.<|im_end|>
<|im_start|>assistant
<Z_142><Z_117><ANSWER>

Question: In a group of 50 children, each child has all their siblings in the group. Each child with no older siblings announces the number of siblings they have, while each child with an older sibling says they have 0 siblings. The average of the numbers everyone announces is \(\frac{12}{25}\). Compute the number of different sets of siblings represented in the group.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Grandma Molly created statues of turtles for her front lawn. The first year, she created 4 statues and placed them on her lawn. The second year, she increased the number of statues on her front lawn by a certain factor. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which she threw away. In the fourth year, she added twice as many new statues as had been broken the year before. At the end of the four years, there were 31 turtle statues on her front lawn. What was the factor by which she increased the number of statues in the second year?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Grandma Molly created statues of turtles for her front lawn. The first year, she created 4 statues and placed them on her lawn. The second year, she increased the number of statues on her front lawn by a certain factor. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which she threw away. In the fourth year, she added twice as many new statues as had been broken the year before. At the end of the four years, there were 31 turtle statues on her front lawn. What was the factor by which she increased the number of statues in the second year?<|im_end|>
<|im_start|>assistant
<Z_477><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Grandma Molly created statues of turtles for her front lawn. The first year, she created 4 statues and placed them on her lawn. The second year, she increased the number of statues on her front lawn by a certain factor. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which she threw away. In the fourth year, she added twice as many new statues as had been broken the year before. At the end of the four years, there were 31 turtle statues on her front lawn. What was the factor by which she increased the number of statues in the second year?<|im_end|>
<|im_start|>assistant
<Z_273><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Grandma Molly created statues of turtles for her front lawn. The first year, she created 4 statues and placed them on her lawn. The second year, she increased the number of statues on her front lawn by a certain factor. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which she threw away. In the fourth year, she added twice as many new statues as had been broken the year before. At the end of the four years, there were 31 turtle statues on her front lawn. What was the factor by which she increased the number of statues in the second year?<|im_end|>
<|im_start|>assistant
<Z_349><Z_45><ANSWER>

Question: Grandma Molly created statues of turtles for her front lawn. The first year, she created 4 statues and placed them on her lawn. The second year, she increased the number of statues on her front lawn by a certain factor. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which she threw away. In the fourth year, she added twice as many new statues as had been broken the year before. At the end of the four years, there were 31 turtle statues on her front lawn. What was the factor by which she increased the number of statues in the second year?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What least number should be added to 1015, so that the sum is completely divisible by a certain number? The answer is 10. What is the divisor?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What least number should be added to 1015, so that the sum is completely divisible by a certain number? The answer is 10. What is the divisor?<|im_end|>
<|im_start|>assistant
<Z_477><Z_445><Z_49><Z_68><Z_97><Z_420><Z_467><Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What least number should be added to 1015, so that the sum is completely divisible by a certain number? The answer is 10. What is the divisor?<|im_end|>
<|im_start|>assistant
<Z_471><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What least number should be added to 1015, so that the sum is completely divisible by a certain number? The answer is 10. What is the divisor?<|im_end|>
<|im_start|>assistant
<Z_328><Z_369><Z_163><Z_444><Z_441><Z_262><Z_8><Z_215><ANSWER>

Question: What least number should be added to 1015, so that the sum is completely divisible by a certain number? The answer is 10. What is the divisor?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Drew prepared 61 servings of butterscotch for her 8 sweet-tooth guests. If the first 3 guests share 2/5 of the servings, the next 4 guests share 1/4 of the servings, and the 8th guest consumes 5 servings, how many servings will be left unconsumed?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Drew prepared 61 servings of butterscotch for her 8 sweet-tooth guests. If the first 3 guests share 2/5 of the servings, the next 4 guests share 1/4 of the servings, and the 8th guest consumes 5 servings, how many servings will be left unconsumed?<|im_end|>
<|im_start|>assistant
<Z_24><Z_328><Z_43><Z_442><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Drew prepared 61 servings of butterscotch for her 8 sweet-tooth guests. If the first 3 guests share 2/5 of the servings, the next 4 guests share 1/4 of the servings, and the 8th guest consumes 5 servings, how many servings will be left unconsumed?<|im_end|>
<|im_start|>assistant
<Z_255><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Drew prepared 61 servings of butterscotch for her 8 sweet-tooth guests. If the first 3 guests share 2/5 of the servings, the next 4 guests share 1/4 of the servings, and the 8th guest consumes 5 servings, how many servings will be left unconsumed?<|im_end|>
<|im_start|>assistant
<Z_141><Z_303><Z_272><Z_78><Z_299><ANSWER>

Question: Drew prepared 61 servings of butterscotch for her 8 sweet-tooth guests. If the first 3 guests share 2/5 of the servings, the next 4 guests share 1/4 of the servings, and the 8th guest consumes 5 servings, how many servings will be left unconsumed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A point $(4\sqrt{3}, d-2)$ is $4d$ units away from the origin. What is the smallest possible value of $d$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A point $(4\sqrt{3}, d-2)$ is $4d$ units away from the origin. What is the smallest possible value of $d$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_217><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A point $(4\sqrt{3}, d-2)$ is $4d$ units away from the origin. What is the smallest possible value of $d$?<|im_end|>
<|im_start|>assistant
<Z_435><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A point $(4\sqrt{3}, d-2)$ is $4d$ units away from the origin. What is the smallest possible value of $d$?<|im_end|>
<|im_start|>assistant
<Z_105><Z_149><ANSWER>

Question: A point $(4\sqrt{3}, d-2)$ is $4d$ units away from the origin. What is the smallest possible value of $d$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the product of $\dbinom{12}{6}$ with the squared value of $\dbinom{5}{2}$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the product of $\dbinom{12}{6}$ with the squared value of $\dbinom{5}{2}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_219><Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the product of $\dbinom{12}{6}$ with the squared value of $\dbinom{5}{2}$.<|im_end|>
<|im_start|>assistant
<Z_313><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the product of $\dbinom{12}{6}$ with the squared value of $\dbinom{5}{2}$.<|im_end|>
<|im_start|>assistant
<Z_448><Z_319><Z_165><ANSWER>

Question: Compute the product of $\dbinom{12}{6}$ with the squared value of $\dbinom{5}{2}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the local values of some digits in the number 2345, if the sum is 2345 and the other digits are 2, 3, and 4?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the local values of some digits in the number 2345, if the sum is 2345 and the other digits are 2, 3, and 4?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the local values of some digits in the number 2345, if the sum is 2345 and the other digits are 2, 3, and 4?<|im_end|>
<|im_start|>assistant
<Z_157><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the local values of some digits in the number 2345, if the sum is 2345 and the other digits are 2, 3, and 4?<|im_end|>
<|im_start|>assistant
<Z_11><ANSWER>

Question: What is the sum of the local values of some digits in the number 2345, if the sum is 2345 and the other digits are 2, 3, and 4?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a $6 \times 8$ table, except the corner ones, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is no more than 10. What is the maximum value that the sum of the numbers in the entire table can take?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a $6 \times 8$ table, except the corner ones, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is no more than 10. What is the maximum value that the sum of the numbers in the entire table can take?<|im_end|>
<|im_start|>assistant
<Z_24><Z_95><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a $6 \times 8$ table, except the corner ones, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is no more than 10. What is the maximum value that the sum of the numbers in the entire table can take?<|im_end|>
<|im_start|>assistant
<Z_325><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a $6 \times 8$ table, except the corner ones, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is no more than 10. What is the maximum value that the sum of the numbers in the entire table can take?<|im_end|>
<|im_start|>assistant
<Z_12><Z_227><ANSWER>

Question: In all the cells of a $6 \times 8$ table, except the corner ones, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is no more than 10. What is the maximum value that the sum of the numbers in the entire table can take?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The time it took car P to travel 800 miles was 2 hours less than the time it took car R to travel the same distance. Car P's average speed was 10 miles per hour greater than that of car R. What was car R's average speed, in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The time it took car P to travel 800 miles was 2 hours less than the time it took car R to travel the same distance. Car P's average speed was 10 miles per hour greater than that of car R. What was car R's average speed, in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_477><Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The time it took car P to travel 800 miles was 2 hours less than the time it took car R to travel the same distance. Car P's average speed was 10 miles per hour greater than that of car R. What was car R's average speed, in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The time it took car P to travel 800 miles was 2 hours less than the time it took car R to travel the same distance. Car P's average speed was 10 miles per hour greater than that of car R. What was car R's average speed, in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_186><Z_51><ANSWER>

Question: The time it took car P to travel 800 miles was 2 hours less than the time it took car R to travel the same distance. Car P's average speed was 10 miles per hour greater than that of car R. What was car R's average speed, in miles per hour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Braelynn has 24 knives, 36 forks, and twice as many teaspoons as knives in her kitchen. She bought 5/12 as many additional knives, 3/8 as many additional forks, and 7/18 as many additional teaspoons. What would be the total number of cutlery pieces she has in her kitchen after the additional purchases?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Braelynn has 24 knives, 36 forks, and twice as many teaspoons as knives in her kitchen. She bought 5/12 as many additional knives, 3/8 as many additional forks, and 7/18 as many additional teaspoons. What would be the total number of cutlery pieces she has in her kitchen after the additional purchases?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Braelynn has 24 knives, 36 forks, and twice as many teaspoons as knives in her kitchen. She bought 5/12 as many additional knives, 3/8 as many additional forks, and 7/18 as many additional teaspoons. What would be the total number of cutlery pieces she has in her kitchen after the additional purchases?<|im_end|>
<|im_start|>assistant
<Z_177><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Braelynn has 24 knives, 36 forks, and twice as many teaspoons as knives in her kitchen. She bought 5/12 as many additional knives, 3/8 as many additional forks, and 7/18 as many additional teaspoons. What would be the total number of cutlery pieces she has in her kitchen after the additional purchases?<|im_end|>
<|im_start|>assistant
<Z_252><ANSWER>

Question: Braelynn has 24 knives, 36 forks, and twice as many teaspoons as knives in her kitchen. She bought 5/12 as many additional knives, 3/8 as many additional forks, and 7/18 as many additional teaspoons. What would be the total number of cutlery pieces she has in her kitchen after the additional purchases?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( c \equiv 25 \pmod{50} \) and \( d \equiv 92 \pmod{50} \), then for what integer \( m \) in the set \{150, 151, 152, ..., 200, 201\} is it true that \( c-d \equiv m \pmod{50} \)?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( c \equiv 25 \pmod{50} \) and \( d \equiv 92 \pmod{50} \), then for what integer \( m \) in the set \{150, 151, 152, ..., 200, 201\} is it true that \( c-d \equiv m \pmod{50} \)?<|im_end|>
<|im_start|>assistant
<Z_24><Z_136><Z_447><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( c \equiv 25 \pmod{50} \) and \( d \equiv 92 \pmod{50} \), then for what integer \( m \) in the set \{150, 151, 152, ..., 200, 201\} is it true that \( c-d \equiv m \pmod{50} \)?<|im_end|>
<|im_start|>assistant
<Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( c \equiv 25 \pmod{50} \) and \( d \equiv 92 \pmod{50} \), then for what integer \( m \) in the set \{150, 151, 152, ..., 200, 201\} is it true that \( c-d \equiv m \pmod{50} \)?<|im_end|>
<|im_start|>assistant
<Z_482><Z_336><Z_130><ANSWER>

Question: If \( c \equiv 25 \pmod{50} \) and \( d \equiv 92 \pmod{50} \), then for what integer \( m \) in the set \{150, 151, 152, ..., 200, 201\} is it true that \( c-d \equiv m \pmod{50} \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p and q are prime numbers, the product p^3 * q^7 has 32 divisors. What is the exponent of p in the product?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p and q are prime numbers, the product p^3 * q^7 has 32 divisors. What is the exponent of p in the product?<|im_end|>
<|im_start|>assistant
<Z_369><Z_238><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p and q are prime numbers, the product p^3 * q^7 has 32 divisors. What is the exponent of p in the product?<|im_end|>
<|im_start|>assistant
<Z_254><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If p and q are prime numbers, the product p^3 * q^7 has 32 divisors. What is the exponent of p in the product?<|im_end|>
<|im_start|>assistant
<Z_71><Z_290><ANSWER>

Question: If p and q are prime numbers, the product p^3 * q^7 has 32 divisors. What is the exponent of p in the product?


<|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 two numbers is 5940, and the greatest common divisor is 22. Given that one of the numbers is 220, what is the other number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least common multiple of two numbers is 5940, and the greatest common divisor is 22. Given that one of the numbers is 220, what is the other number?<|im_end|>
<|im_start|>assistant
<Z_24><Z_454><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 two numbers is 5940, and the greatest common divisor is 22. Given that one of the numbers is 220, what is the other number?<|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
The least common multiple of two numbers is 5940, and the greatest common divisor is 22. Given that one of the numbers is 220, what is the other number?<|im_end|>
<|im_start|>assistant
<Z_404><Z_203><ANSWER>

Question: The least common multiple of two numbers is 5940, and the greatest common divisor is 22. Given that one of the numbers is 220, what is the other number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the function $f(x)=x^2+ax-1$ is an even function, then find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the function $f(x)=x^2+ax-1$ is an even function, then find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the function $f(x)=x^2+ax-1$ is an even function, then find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_314><ANSWER>

Question: If the function $f(x)=x^2+ax-1$ is an even function, then find the value of $a$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Jacksonville Walmart normally gets 120 customer complaints per day. That number increases by 1/3rd when they're short-staffed and this increases by another 20% when the self-checkout is broken. The store got 576 complaints during a certain number of days when it was short-staffed and the self-checkout was broken. How many days was the store short-staffed and the self-checkout broken?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Jacksonville Walmart normally gets 120 customer complaints per day. That number increases by 1/3rd when they're short-staffed and this increases by another 20% when the self-checkout is broken. The store got 576 complaints during a certain number of days when it was short-staffed and the self-checkout was broken. How many days was the store short-staffed and the self-checkout broken?<|im_end|>
<|im_start|>assistant
<Z_369><Z_75><Z_215><Z_153><Z_425><Z_244><Z_467><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Jacksonville Walmart normally gets 120 customer complaints per day. That number increases by 1/3rd when they're short-staffed and this increases by another 20% when the self-checkout is broken. The store got 576 complaints during a certain number of days when it was short-staffed and the self-checkout was broken. How many days was the store short-staffed and the self-checkout broken?<|im_end|>
<|im_start|>assistant
<Z_444><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Jacksonville Walmart normally gets 120 customer complaints per day. That number increases by 1/3rd when they're short-staffed and this increases by another 20% when the self-checkout is broken. The store got 576 complaints during a certain number of days when it was short-staffed and the self-checkout was broken. How many days was the store short-staffed and the self-checkout broken?<|im_end|>
<|im_start|>assistant
<Z_404><Z_299><Z_430><Z_319><Z_261><Z_214><Z_57><ANSWER>

Question: The Jacksonville Walmart normally gets 120 customer complaints per day. That number increases by 1/3rd when they're short-staffed and this increases by another 20% when the self-checkout is broken. The store got 576 complaints during a certain number of days when it was short-staffed and the self-checkout was broken. How many days was the store short-staffed and the self-checkout broken?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|im_end|>
<|im_start|>assistant
<Z_241><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?<|im_end|>
<|im_start|>assistant
<Z_45><ANSWER>

Question: Porche has some time to get all her homework done. Her math homework takes her 45 minutes. Her English homework takes her 30 minutes. Her science homework takes her 50 minutes. Her history homework takes her 25 minutes. She also has a special project due the next day. She has 30 minutes left to get that project done. How much time did Porche initially have to get all her homework done?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_113><Z_443><Z_168><Z_148><Z_252><Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_299><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).<|im_end|>
<|im_start|>assistant
<Z_473><Z_239><Z_365><Z_241><Z_1><Z_227><Z_7><ANSWER>

Question: Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_477><Z_43><Z_187><Z_185><Z_205><Z_200><Z_437><Z_219><Z_214><Z_161><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 possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_500><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 possible three-digit number that is divisible by 3, 6, and 4?<|im_end|>
<|im_start|>assistant
<Z_347><Z_487><Z_497><Z_437><Z_511><Z_17><Z_276><Z_291><Z_30><Z_15><ANSWER>

Question: What is the greatest possible three-digit number that is divisible by 3, 6, and 4?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_369><Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).<|im_end|>
<|im_start|>assistant
<Z_52><Z_28><ANSWER>

Question:
Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_369><Z_354><Z_155><Z_275><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_210><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.<|im_end|>
<|im_start|>assistant
<Z_118><Z_69><Z_56><Z_7><ANSWER>

Question: Six people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is to be determined.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_24><Z_112><Z_187><Z_454><Z_225><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?<|im_end|>
<|im_start|>assistant
<Z_390><Z_475><Z_309><Z_313><Z_171><ANSWER>

Question: How many three-digit numbers are there in which the hundreds digit is greater than the tens digit, and the tens digit is greater than the ones digit?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_369><Z_502><Z_503><Z_461><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_382><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?<|im_end|>
<|im_start|>assistant
<Z_134><Z_41><Z_271><Z_232><ANSWER>

Question: A fills a tank in some minutes, and B fills the same tank in 20 mins, and C empties it in 40 mins. When A, B, and C work together, the tank takes 17.14285714285714 minutes to fill. How much time does A take to fill the tank alone?