<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divisible by 9 if the sum of its digits is divisible by 9. $C\,498\,DF4$ is divisible by 9 where $C, D,$ and $F$ each represent a single digit. What is the sum of all possible values of $C+D+F$?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divisible by 9 if the sum of its digits is divisible by 9. $C\,498\,DF4$ is divisible by 9 where $C, D,$ and $F$ each represent a single digit. What is the sum of all possible values of $C+D+F$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_106><Z_37><Z_121><Z_99><Z_75><Z_9><Z_8><Z_66><Z_34><Z_16><Z_90><Z_16><Z_90><Z_93><Z_118><Z_59><Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divisible by 9 if the sum of its digits is divisible by 9. $C\,498\,DF4$ is divisible by 9 where $C, D,$ and $F$ each represent a single digit. What is the sum of all possible values of $C+D+F$?<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divisible by 9 if the sum of its digits is divisible by 9. $C\,498\,DF4$ is divisible by 9 where $C, D,$ and $F$ each represent a single digit. What is the sum of all possible values of $C+D+F$?<|im_end|>
<|im_start|>assistant
<Z_42><Z_116><Z_51><Z_46><Z_2><Z_34><Z_49><Z_61><Z_103><Z_104><Z_17><Z_89><Z_75><Z_78><Z_116><Z_88><Z_24><Z_88><ANSWER>

Question: A number is divisible by 9 if the sum of its digits is divisible by 9. $C\,498\,DF4$ is divisible by 9 where $C, D,$ and $F$ each represent a single digit. What is the sum of all possible values of $C+D+F$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 50 positive even integers is 2550. What is the sum of even integers from a certain number to 200 inclusive if it is 7550?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 50 positive even integers is 2550. What is the sum of even integers from a certain number to 200 inclusive if it is 7550?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 50 positive even integers is 2550. What is the sum of even integers from a certain number to 200 inclusive if it is 7550?<|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 sum of the first 50 positive even integers is 2550. What is the sum of even integers from a certain number to 200 inclusive if it is 7550?<|im_end|>
<|im_start|>assistant
<Z_112><ANSWER>

Question: The sum of the first 50 positive even integers is 2550. What is the sum of even integers from a certain number to 200 inclusive if it is 7550?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last week, the Smart Mart sold 45 science kits. This week, they increased their sales and sold 20% more science kits. They also sold puzzles, but the number of puzzles was 30% less than the science kits sold this week. How many puzzles did the Smart Mart sell this week?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last week, the Smart Mart sold 45 science kits. This week, they increased their sales and sold 20% more science kits. They also sold puzzles, but the number of puzzles was 30% less than the science kits sold this week. How many puzzles did the Smart Mart sell this week?<|im_end|>
<|im_start|>assistant
<Z_94><Z_40><Z_86><Z_107><Z_40><Z_62><Z_49><Z_115><Z_34><Z_71><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last week, the Smart Mart sold 45 science kits. This week, they increased their sales and sold 20% more science kits. They also sold puzzles, but the number of puzzles was 30% less than the science kits sold this week. How many puzzles did the Smart Mart sell this week?<|im_end|>
<|im_start|>assistant
<Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Last week, the Smart Mart sold 45 science kits. This week, they increased their sales and sold 20% more science kits. They also sold puzzles, but the number of puzzles was 30% less than the science kits sold this week. How many puzzles did the Smart Mart sell this week?<|im_end|>
<|im_start|>assistant
<Z_108><Z_32><Z_44><Z_65><Z_10><Z_65><Z_118><Z_87><Z_26><Z_66><Z_9><ANSWER>

Question: Last week, the Smart Mart sold 45 science kits. This week, they increased their sales and sold 20% more science kits. They also sold puzzles, but the number of puzzles was 30% less than the science kits sold this week. How many puzzles did the Smart Mart sell this week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A math conference is organizing a series with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, requiring Dr. Smith to be scheduled after Dr. Jones. Additionally, Dr. Green's lecture depends on Dr. Brown's lecture, requiring Dr. Green to be scheduled after Dr. Brown. In how many orders can the seven lecturers be scheduled?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A math conference is organizing a series with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, requiring Dr. Smith to be scheduled after Dr. Jones. Additionally, Dr. Green's lecture depends on Dr. Brown's lecture, requiring Dr. Green to be scheduled after Dr. Brown. In how many orders can the seven lecturers be scheduled?<|im_end|>
<|im_start|>assistant
<Z_104><Z_91><Z_96><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A math conference is organizing a series with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, requiring Dr. Smith to be scheduled after Dr. Jones. Additionally, Dr. Green's lecture depends on Dr. Brown's lecture, requiring Dr. Green to be scheduled after Dr. Brown. In how many orders can the seven lecturers be scheduled?<|im_end|>
<|im_start|>assistant
<Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A math conference is organizing a series with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, requiring Dr. Smith to be scheduled after Dr. Jones. Additionally, Dr. Green's lecture depends on Dr. Brown's lecture, requiring Dr. Green to be scheduled after Dr. Brown. In how many orders can the seven lecturers be scheduled?<|im_end|>
<|im_start|>assistant
<Z_33><Z_105><Z_26><Z_104><ANSWER>

Question: A math conference is organizing a series with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, requiring Dr. Smith to be scheduled after Dr. Jones. Additionally, Dr. Green's lecture depends on Dr. Brown's lecture, requiring Dr. Green to be scheduled after Dr. Brown. In how many orders can the seven lecturers be scheduled?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A$ be the product of the divisors of $60$. How many distinct prime factors does $A$ have?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A$ be the product of the divisors of $60$. How many distinct prime factors does $A$ have?<|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
Let $A$ be the product of the divisors of $60$. How many distinct prime factors does $A$ have?<|im_end|>
<|im_start|>assistant
<Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A$ be the product of the divisors of $60$. How many distinct prime factors does $A$ have?<|im_end|>
<|im_start|>assistant
<Z_11><ANSWER>

Question: Let $A$ be the product of the divisors of $60$. How many distinct prime factors does $A$ have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a right-angled triangle with one side of length 11 and the other two sides being natural numbers, what is the perimeter of this triangle?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a right-angled triangle with one side of length 11 and the other two sides being natural numbers, what is the perimeter of this triangle?<|im_end|>
<|im_start|>assistant
<Z_94><Z_86><Z_87><Z_97><Z_87><Z_65><Z_80><Z_126><Z_12><Z_49><Z_11><Z_16><Z_63><Z_103><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a right-angled triangle with one side of length 11 and the other two sides being natural numbers, what is the perimeter of this triangle?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a right-angled triangle with one side of length 11 and the other two sides being natural numbers, what is the perimeter of this triangle?<|im_end|>
<|im_start|>assistant
<Z_113><Z_123><Z_80><Z_37><Z_33><Z_43><Z_78><Z_18><Z_75><Z_123><Z_29><Z_10><Z_105><Z_44><Z_27><ANSWER>

Question: Given a right-angled triangle with one side of length 11 and the other two sides being natural numbers, what is the perimeter of this triangle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average marks of a class of 24 students is 40 and that of another class of 50 students is a certain value. The average marks of all the students is 53.513513513513516. What is the average marks of the second class?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average marks of a class of 24 students is 40 and that of another class of 50 students is a certain value. The average marks of all the students is 53.513513513513516. What is the average marks of the second class?<|im_end|>
<|im_start|>assistant
<Z_57><Z_38><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average marks of a class of 24 students is 40 and that of another class of 50 students is a certain value. The average marks of all the students is 53.513513513513516. What is the average marks of the second class?<|im_end|>
<|im_start|>assistant
<Z_7><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average marks of a class of 24 students is 40 and that of another class of 50 students is a certain value. The average marks of all the students is 53.513513513513516. What is the average marks of the second class?<|im_end|>
<|im_start|>assistant
<Z_80><Z_29><Z_67><ANSWER>

Question: The average marks of a class of 24 students is 40 and that of another class of 50 students is a certain value. The average marks of all the students is 53.513513513513516. What is the average marks of the second class?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a **smart integer** as an even integer greater than 50, less than 200, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a **smart integer** as an even integer greater than 50, less than 200, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_104><Z_73><Z_47><Z_100><Z_31><Z_99><Z_78><Z_98><Z_64><Z_91><Z_78><Z_55><Z_117><Z_49><Z_20><Z_126><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a **smart integer** as an even integer greater than 50, less than 200, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a **smart integer** as an even integer greater than 50, less than 200, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_42><Z_23><Z_44><Z_95><Z_24><Z_107><Z_122><Z_28><Z_61><Z_107><Z_58><Z_109><Z_18><Z_93><Z_72><Z_14><Z_104><ANSWER>

Question: Define a **smart integer** as an even integer greater than 50, less than 200, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a fellow borrowed a certain sum of money at 6 % per annum at simple interest and in 6 years the interest amounted to rs . 672 less than the sum lent . what was the sum lent ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a fellow borrowed a certain sum of money at 6 % per annum at simple interest and in 6 years the interest amounted to rs . 672 less than the sum lent . what was the sum lent ?<|im_end|>
<|im_start|>assistant
<Z_104><Z_124><Z_22><Z_119><Z_19><Z_63><Z_45><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a fellow borrowed a certain sum of money at 6 % per annum at simple interest and in 6 years the interest amounted to rs . 672 less than the sum lent . what was the sum lent ?<|im_end|>
<|im_start|>assistant
<Z_8><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a fellow borrowed a certain sum of money at 6 % per annum at simple interest and in 6 years the interest amounted to rs . 672 less than the sum lent . what was the sum lent ?<|im_end|>
<|im_start|>assistant
<Z_108><Z_37><Z_65><Z_11><Z_33><Z_67><Z_100><Z_40><ANSWER>

Question: a fellow borrowed a certain sum of money at 6 % per annum at simple interest and in 6 years the interest amounted to rs . 672 less than the sum lent . what was the sum lent ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram below, triangle $PQR$ has been reflected over line segment $\overline{PR}$ to produce triangle $PQ'S'$. If $PQ = 9$, $QR = 15$, and $PS = 20$, then find the length of $PQ'$.

[asy]
size(250);
pair P,Q,R,S,PP,QQ,SS;
P = (0,0);
Q = (9,0);
R = (4,12);
S = (20,0);
QQ = reflect(P,R)*Q;
SS = reflect(P,R)*S;
PP = P;
draw(P--Q--R--P--S--R);
draw(P--QQ--SS--P);
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$S$",S,SE);
label("$Q'$",QQ,NW);
label("$S'$",SS,NE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram below, triangle $PQR$ has been reflected over line segment $\overline{PR}$ to produce triangle $PQ'S'$. If $PQ = 9$, $QR = 15$, and $PS = 20$, then find the length of $PQ'$.

[asy]
size(250);
pair P,Q,R,S,PP,QQ,SS;
P = (0,0);
Q = (9,0);
R = (4,12);
S = (20,0);
QQ = reflect(P,R)*Q;
SS = reflect(P,R)*S;
PP = P;
draw(P--Q--R--P--S--R);
draw(P--QQ--SS--P);
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$S$",S,SE);
label("$Q'$",QQ,NW);
label("$S'$",SS,NE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_94><Z_71><Z_110><Z_16><Z_91><Z_92><Z_70><Z_91><Z_39><Z_9><Z_66><Z_119><Z_16><Z_76><Z_12><Z_91><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram below, triangle $PQR$ has been reflected over line segment $\overline{PR}$ to produce triangle $PQ'S'$. If $PQ = 9$, $QR = 15$, and $PS = 20$, then find the length of $PQ'$.

[asy]
size(250);
pair P,Q,R,S,PP,QQ,SS;
P = (0,0);
Q = (9,0);
R = (4,12);
S = (20,0);
QQ = reflect(P,R)*Q;
SS = reflect(P,R)*S;
PP = P;
draw(P--Q--R--P--S--R);
draw(P--QQ--SS--P);
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$S$",S,SE);
label("$Q'$",QQ,NW);
label("$S'$",SS,NE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram below, triangle $PQR$ has been reflected over line segment $\overline{PR}$ to produce triangle $PQ'S'$. If $PQ = 9$, $QR = 15$, and $PS = 20$, then find the length of $PQ'$.

[asy]
size(250);
pair P,Q,R,S,PP,QQ,SS;
P = (0,0);
Q = (9,0);
R = (4,12);
S = (20,0);
QQ = reflect(P,R)*Q;
SS = reflect(P,R)*S;
PP = P;
draw(P--Q--R--P--S--R);
draw(P--QQ--SS--P);
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$S$",S,SE);
label("$Q'$",QQ,NW);
label("$S'$",SS,NE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_8><Z_124><Z_27><Z_34><Z_94><Z_15><Z_59><Z_66><Z_83><Z_33><Z_2><Z_91><Z_105><Z_48><Z_122><Z_56><ANSWER>

Question:
In the diagram below, triangle $PQR$ has been reflected over line segment $\overline{PR}$ to produce triangle $PQ'S'$. If $PQ = 9$, $QR = 15$, and $PS = 20$, then find the length of $PQ'$.

[asy]
size(250);
pair P,Q,R,S,PP,QQ,SS;
P = (0,0);
Q = (9,0);
R = (4,12);
S = (20,0);
QQ = reflect(P,R)*Q;
SS = reflect(P,R)*S;
PP = P;
draw(P--Q--R--P--S--R);
draw(P--QQ--SS--P);
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$S$",S,SE);
label("$Q'$",QQ,NW);
label("$S'$",SS,NE);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kylie makes some beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. 20 beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beaded necklaces did Kylie make on Monday?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kylie makes some beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. 20 beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beaded necklaces did Kylie make on Monday?<|im_end|>
<|im_start|>assistant
<Z_113><Z_50><Z_114><Z_93><Z_108><Z_43><Z_73><Z_101><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kylie makes some beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. 20 beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beaded necklaces did Kylie make on Monday?<|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
Kylie makes some beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. 20 beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beaded necklaces did Kylie make on Monday?<|im_end|>
<|im_start|>assistant
<Z_74><Z_66><Z_86><Z_51><Z_23><Z_101><Z_31><Z_89><Z_63><ANSWER>

Question: Kylie makes some beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. 20 beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beaded necklaces did Kylie make on Monday?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.<|im_end|>
<|im_start|>assistant
<Z_104><Z_52><Z_77><Z_40><Z_70><Z_54><Z_20><Z_38><Z_67><Z_5><Z_93><Z_26><Z_96><Z_118><Z_33><Z_5><Z_78><Z_47><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.<|im_end|>
<|im_start|>assistant
<Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.<|im_end|>
<|im_start|>assistant
<Z_125><Z_126><Z_28><Z_105><Z_4><Z_59><Z_67><Z_60><Z_81><Z_22><Z_23><Z_39><Z_77><Z_90><Z_35><Z_107><Z_122><Z_77><Z_114><ANSWER>

Question: 27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the number of diagonals in a convex polygon that has 25 sides.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the number of diagonals in a convex polygon that has 25 sides.<|im_end|>
<|im_start|>assistant
<Z_57><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the number of diagonals in a convex polygon that has 25 sides.<|im_end|>
<|im_start|>assistant
<Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the number of diagonals in a convex polygon that has 25 sides.<|im_end|>
<|im_start|>assistant
<Z_46><Z_116><ANSWER>

Question: Calculate the number of diagonals in a convex polygon that has 25 sides.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if | x - 5 | = 23 what is the sum of all the values of x .<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if | x - 5 | = 23 what is the sum of all the values of x .<|im_end|>
<|im_start|>assistant
<Z_57><Z_81><Z_36><Z_59><Z_93><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if | x - 5 | = 23 what is the sum of all the values of x .<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if | x - 5 | = 23 what is the sum of all the values of x .<|im_end|>
<|im_start|>assistant
<Z_91><Z_110><Z_95><Z_127><Z_113><Z_59><ANSWER>

Question: if | x - 5 | = 23 what is the sum of all the values of x .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dragon selects three positive real numbers with sum  $100$ , uniformly at random.  He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription.  If the probability the three new numbers still sum to  $100$  is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $100m+n$ .

*Proposed by Aaron Lin*<|im_end|>
<|im_start|>assistant
<Z_104><Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dragon selects three positive real numbers with sum  $100$ , uniformly at random.  He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription.  If the probability the three new numbers still sum to  $100$  is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $100m+n$ .

*Proposed by Aaron Lin*<|im_end|>
<|im_start|>assistant
<Z_94><Z_81><Z_99><Z_38><Z_83><Z_53><Z_115><Z_33><Z_125><Z_117><Z_66><Z_90><Z_79><Z_103><Z_78><Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dragon selects three positive real numbers with sum  $100$ , uniformly at random.  He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription.  If the probability the three new numbers still sum to  $100$  is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $100m+n$ .

*Proposed by Aaron Lin*<|im_end|>
<|im_start|>assistant
<Z_91><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dragon selects three positive real numbers with sum  $100$ , uniformly at random.  He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription.  If the probability the three new numbers still sum to  $100$  is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $100m+n$ .

*Proposed by Aaron Lin*<|im_end|>
<|im_start|>assistant
<Z_71><Z_41><Z_123><Z_126><Z_7><Z_6><Z_46><Z_127><Z_11><Z_34><Z_127><Z_107><Z_82><Z_34><Z_11><Z_53><ANSWER>

Question:
Dragon selects three positive real numbers with sum  $100$ , uniformly at random.  He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription.  If the probability the three new numbers still sum to  $100$  is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $100m+n$ .

*Proposed by Aaron Lin*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares less than 2500 have a ones digit of 4, 5, or 6?<|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
How many perfect squares less than 2500 have a ones digit of 4, 5, or 6?<|im_end|>
<|im_start|>assistant
<Z_94><Z_87><Z_70><Z_22><Z_121><Z_47><Z_66><Z_103><Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares less than 2500 have a ones digit of 4, 5, or 6?<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many perfect squares less than 2500 have a ones digit of 4, 5, or 6?<|im_end|>
<|im_start|>assistant
<Z_41><Z_100><Z_105><Z_78><Z_40><Z_8><Z_64><Z_110><Z_87><ANSWER>

Question: How many perfect squares less than 2500 have a ones digit of 4, 5, or 6?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A trapezoid is 8 centimeters (cm) high, the length of the top side is 6 centimeters (cm) shorter than the length of the bottom side, and it is 72 square centimeters (㎠) wide. What is the length of the top side of this trapezoid?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A trapezoid is 8 centimeters (cm) high, the length of the top side is 6 centimeters (cm) shorter than the length of the bottom side, and it is 72 square centimeters (㎠) wide. What is the length of the top side of this trapezoid?<|im_end|>
<|im_start|>assistant
<Z_104><Z_30><Z_47><Z_80><Z_33><Z_75><Z_79><Z_55><Z_126><Z_61><Z_88><Z_28><Z_96><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A trapezoid is 8 centimeters (cm) high, the length of the top side is 6 centimeters (cm) shorter than the length of the bottom side, and it is 72 square centimeters (㎠) wide. What is the length of the top side of this trapezoid?<|im_end|>
<|im_start|>assistant
<Z_112><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A trapezoid is 8 centimeters (cm) high, the length of the top side is 6 centimeters (cm) shorter than the length of the bottom side, and it is 72 square centimeters (㎠) wide. What is the length of the top side of this trapezoid?<|im_end|>
<|im_start|>assistant
<Z_60><Z_118><Z_24><Z_38><Z_22><Z_68><Z_10><Z_16><Z_31><Z_14><Z_27><Z_36><Z_110><Z_62><ANSWER>

Question: A trapezoid is 8 centimeters (cm) high, the length of the top side is 6 centimeters (cm) shorter than the length of the bottom side, and it is 72 square centimeters (㎠) wide. What is the length of the top side of this trapezoid?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A field is 90 metre long and some metres broad. A 25 metre long, 20 metre broad and 4 metre deep tank is dug in the field and the earth taken out is spread evenly over the remaining field. The level of the field will rise by 0.5 metre. How broad is the field?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A field is 90 metre long and some metres broad. A 25 metre long, 20 metre broad and 4 metre deep tank is dug in the field and the earth taken out is spread evenly over the remaining field. The level of the field will rise by 0.5 metre. How broad is the field?<|im_end|>
<|im_start|>assistant
<Z_94><Z_30><Z_99><Z_50><Z_75><Z_80><Z_56><Z_26><Z_87><Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A field is 90 metre long and some metres broad. A 25 metre long, 20 metre broad and 4 metre deep tank is dug in the field and the earth taken out is spread evenly over the remaining field. The level of the field will rise by 0.5 metre. How broad is the field?<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A field is 90 metre long and some metres broad. A 25 metre long, 20 metre broad and 4 metre deep tank is dug in the field and the earth taken out is spread evenly over the remaining field. The level of the field will rise by 0.5 metre. How broad is the field?<|im_end|>
<|im_start|>assistant
<Z_65><Z_4><Z_88><Z_88><Z_104><Z_19><Z_69><Z_0><Z_55><Z_95><ANSWER>

Question: A field is 90 metre long and some metres broad. A 25 metre long, 20 metre broad and 4 metre deep tank is dug in the field and the earth taken out is spread evenly over the remaining field. The level of the field will rise by 0.5 metre. How broad is the field?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an athletics team, there are 42 male athletes and 30 female athletes. A stratified sampling method is used to draw a sample of size $n$ from all the athletes. If the sample includes 5 female athletes, then the value of $n$ is ____.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an athletics team, there are 42 male athletes and 30 female athletes. A stratified sampling method is used to draw a sample of size $n$ from all the athletes. If the sample includes 5 female athletes, then the value of $n$ is ____.<|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
In an athletics team, there are 42 male athletes and 30 female athletes. A stratified sampling method is used to draw a sample of size $n$ from all the athletes. If the sample includes 5 female athletes, then the value of $n$ is ____.<|im_end|>
<|im_start|>assistant
<Z_85><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an athletics team, there are 42 male athletes and 30 female athletes. A stratified sampling method is used to draw a sample of size $n$ from all the athletes. If the sample includes 5 female athletes, then the value of $n$ is ____.<|im_end|>
<|im_start|>assistant
<Z_4><ANSWER>

Question: In an athletics team, there are 42 male athletes and 30 female athletes. A stratified sampling method is used to draw a sample of size $n$ from all the athletes. If the sample includes 5 female athletes, then the value of $n$ is ____.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 3 when divided by 5?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 3 when divided by 5?<|im_end|>
<|im_start|>assistant
<Z_94><Z_59><Z_43><Z_80><Z_121><Z_16><Z_78><Z_54><Z_26><Z_58><Z_59><Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 3 when divided by 5?<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 3 when divided by 5?<|im_end|>
<|im_start|>assistant
<Z_94><Z_127><Z_88><Z_90><Z_67><Z_3><Z_38><Z_105><Z_88><Z_125><Z_70><Z_107><ANSWER>

Question: What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 3 when divided by 5?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonempty subsets of {1, 2, 3, ..., 12} have the property that the sum of the largest element and the smallest element is 13?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonempty subsets of {1, 2, 3, ..., 12} have the property that the sum of the largest element and the smallest element is 13?<|im_end|>
<|im_start|>assistant
<Z_57><Z_72><Z_88><Z_12><Z_95><Z_48><Z_59><Z_126><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonempty subsets of {1, 2, 3, ..., 12} have the property that the sum of the largest element and the smallest element is 13?<|im_end|>
<|im_start|>assistant
<Z_89><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonempty subsets of {1, 2, 3, ..., 12} have the property that the sum of the largest element and the smallest element is 13?<|im_end|>
<|im_start|>assistant
<Z_102><Z_26><Z_86><Z_27><Z_95><Z_66><Z_20><Z_24><Z_67><ANSWER>

Question: How many nonempty subsets of {1, 2, 3, ..., 12} have the property that the sum of the largest element and the smallest element is 13?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the polar coordinate system, the polar coordinates of two points $A$ and $B$ are given as $(6, \frac{\pi}{3})$ and $(4, \frac{\pi}{6})$, respectively. Calculate the area of $\triangle AOB$, where $O$ is the pole.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the polar coordinate system, the polar coordinates of two points $A$ and $B$ are given as $(6, \frac{\pi}{3})$ and $(4, \frac{\pi}{6})$, respectively. Calculate the area of $\triangle AOB$, where $O$ is the pole.<|im_end|>
<|im_start|>assistant
<Z_104><Z_86><Z_73><Z_68><Z_126><Z_115><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the polar coordinate system, the polar coordinates of two points $A$ and $B$ are given as $(6, \frac{\pi}{3})$ and $(4, \frac{\pi}{6})$, respectively. Calculate the area of $\triangle AOB$, where $O$ is the pole.<|im_end|>
<|im_start|>assistant
<Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the polar coordinate system, the polar coordinates of two points $A$ and $B$ are given as $(6, \frac{\pi}{3})$ and $(4, \frac{\pi}{6})$, respectively. Calculate the area of $\triangle AOB$, where $O$ is the pole.<|im_end|>
<|im_start|>assistant
<Z_12><Z_65><Z_85><Z_103><Z_118><Z_11><Z_62><ANSWER>

Question: In the polar coordinate system, the polar coordinates of two points $A$ and $B$ are given as $(6, \frac{\pi}{3})$ and $(4, \frac{\pi}{6})$, respectively. Calculate the area of $\triangle AOB$, where $O$ is the pole.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the perimeter of a sector of a circle is $16$, find the radius and central angle of the sector when its area is at its maximum. Also, find the maximum area.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the perimeter of a sector of a circle is $16$, find the radius and central angle of the sector when its area is at its maximum. Also, find the maximum area.<|im_end|>
<|im_start|>assistant
<Z_113><Z_71><Z_115><Z_91><Z_63><Z_32><Z_87><Z_59><Z_63><Z_75><Z_79><Z_78><Z_96><Z_125><Z_80><Z_75><Z_95><Z_96><Z_16><Z_66><Z_52><Z_26><Z_92><Z_99><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the perimeter of a sector of a circle is $16$, find the radius and central angle of the sector when its area is at its maximum. Also, find the maximum area.<|im_end|>
<|im_start|>assistant
<Z_42><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the perimeter of a sector of a circle is $16$, find the radius and central angle of the sector when its area is at its maximum. Also, find the maximum area.<|im_end|>
<|im_start|>assistant
<Z_22><Z_125><Z_36><Z_76><Z_46><Z_0><Z_14><Z_73><Z_57><Z_45><Z_60><Z_56><Z_9><Z_113><Z_118><Z_23><Z_55><Z_105><Z_61><Z_118><Z_82><Z_38><Z_27><Z_85><Z_83><ANSWER>

Question: Given that the perimeter of a sector of a circle is $16$, find the radius and central angle of the sector when its area is at its maximum. Also, find the maximum area.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation | x / some_number | + | y / some_number | encloses a certain region on the coordinate plane. The area of this region is 32. What is the value of some_number?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation | x / some_number | + | y / some_number | encloses a certain region on the coordinate plane. The area of this region is 32. What is the value of some_number?<|im_end|>
<|im_start|>assistant
<Z_57><Z_78><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation | x / some_number | + | y / some_number | encloses a certain region on the coordinate plane. The area of this region is 32. What is the value of some_number?<|im_end|>
<|im_start|>assistant
<Z_6><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation | x / some_number | + | y / some_number | encloses a certain region on the coordinate plane. The area of this region is 32. What is the value of some_number?<|im_end|>
<|im_start|>assistant
<Z_10><Z_71><Z_12><ANSWER>

Question: The equation | x / some_number | + | y / some_number | encloses a certain region on the coordinate plane. The area of this region is 32. What is the value of some_number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?<|im_end|>
<|im_start|>assistant
<Z_104><Z_38><Z_49><Z_30><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?<|im_end|>
<|im_start|>assistant
<Z_126><Z_97><Z_71><Z_61><Z_114><ANSWER>

Question: In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?


<|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 prime numbers between 20 and 40?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the prime numbers between 20 and 40?<|im_end|>
<|im_start|>assistant
<Z_104><Z_115><Z_82><Z_34><Z_119><Z_125><Z_126><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 prime numbers between 20 and 40?<|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
What is the sum of the prime numbers between 20 and 40?<|im_end|>
<|im_start|>assistant
<Z_114><Z_70><Z_69><Z_104><Z_107><Z_2><Z_26><ANSWER>

Question: What is the sum of the prime numbers between 20 and 40?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The three angles of a triangle add up to 180 degrees. The largest angle is 5 times bigger than the smallest, and the middle angle is 3 times bigger than the smallest. How many degrees is the smallest angle?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The three angles of a triangle add up to 180 degrees. The largest angle is 5 times bigger than the smallest, and the middle angle is 3 times bigger than the smallest. How many degrees is the smallest angle?<|im_end|>
<|im_start|>assistant
<Z_94><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The three angles of a triangle add up to 180 degrees. The largest angle is 5 times bigger than the smallest, and the middle angle is 3 times bigger than the smallest. How many degrees is the smallest angle?<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The three angles of a triangle add up to 180 degrees. The largest angle is 5 times bigger than the smallest, and the middle angle is 3 times bigger than the smallest. How many degrees is the smallest angle?<|im_end|>
<|im_start|>assistant
<Z_4><Z_0><ANSWER>

Question: The three angles of a triangle add up to 180 degrees. The largest angle is 5 times bigger than the smallest, and the middle angle is 3 times bigger than the smallest. How many degrees is the smallest angle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?<|im_end|>
<|im_start|>assistant
<Z_30><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?<|im_end|>
<|im_start|>assistant
<Z_111><ANSWER>

Question: In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving parallelly on adjacent tracks. The first train crosses a stationary second train (length = 180m) in 18 seconds and crosses a platform of length 250m in 24 seconds, while the second train crosses a platform of 200m in 22 seconds. Assuming both trains move at constant speeds throughout, find the length of the first train.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving parallelly on adjacent tracks. The first train crosses a stationary second train (length = 180m) in 18 seconds and crosses a platform of length 250m in 24 seconds, while the second train crosses a platform of 200m in 22 seconds. Assuming both trains move at constant speeds throughout, find the length of the first train.<|im_end|>
<|im_start|>assistant
<Z_113><Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving parallelly on adjacent tracks. The first train crosses a stationary second train (length = 180m) in 18 seconds and crosses a platform of length 250m in 24 seconds, while the second train crosses a platform of 200m in 22 seconds. Assuming both trains move at constant speeds throughout, find the length of the first train.<|im_end|>
<|im_start|>assistant
<Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving parallelly on adjacent tracks. The first train crosses a stationary second train (length = 180m) in 18 seconds and crosses a platform of length 250m in 24 seconds, while the second train crosses a platform of 200m in 22 seconds. Assuming both trains move at constant speeds throughout, find the length of the first train.<|im_end|>
<|im_start|>assistant
<Z_116><Z_46><ANSWER>

Question: Two trains are moving parallelly on adjacent tracks. The first train crosses a stationary second train (length = 180m) in 18 seconds and crosses a platform of length 250m in 24 seconds, while the second train crosses a platform of 200m in 22 seconds. Assuming both trains move at constant speeds throughout, find the length of the first train.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let triangle $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ closest to $A$, such that lines $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{3}{7}$, find $\tan C$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let triangle $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ closest to $A$, such that lines $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{3}{7}$, find $\tan C$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_108><Z_91><Z_98><Z_75><Z_87><Z_11><Z_58><Z_103><Z_87><Z_118><Z_87><Z_19><Z_76><Z_87><Z_71><Z_117><Z_88><Z_39><Z_87><Z_67><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let triangle $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ closest to $A$, such that lines $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{3}{7}$, find $\tan C$.<|im_end|>
<|im_start|>assistant
<Z_7><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let triangle $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ closest to $A$, such that lines $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{3}{7}$, find $\tan C$.<|im_end|>
<|im_start|>assistant
<Z_27><Z_60><Z_44><Z_2><Z_22><Z_82><Z_87><Z_84><Z_13><Z_61><Z_87><Z_107><Z_112><Z_115><Z_46><Z_83><Z_43><Z_22><Z_8><Z_91><Z_73><Z_29><ANSWER>

Question: Let triangle $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ closest to $A$, such that lines $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{3}{7}$, find $\tan C$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabel had 2 pages of math homework and 4 pages of reading homework. If each page had a certain number of problems on it, she had to complete 30 problems total. How many problems were on each page?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabel had 2 pages of math homework and 4 pages of reading homework. If each page had a certain number of problems on it, she had to complete 30 problems total. How many problems were on each page?<|im_end|>
<|im_start|>assistant
<Z_94><Z_100><Z_100><Z_125><Z_86><Z_48><Z_99><Z_52><Z_126><Z_54><Z_96><Z_56><Z_126><Z_52><Z_17><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabel had 2 pages of math homework and 4 pages of reading homework. If each page had a certain number of problems on it, she had to complete 30 problems total. How many problems were on each page?<|im_end|>
<|im_start|>assistant
<Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isabel had 2 pages of math homework and 4 pages of reading homework. If each page had a certain number of problems on it, she had to complete 30 problems total. How many problems were on each page?<|im_end|>
<|im_start|>assistant
<Z_11><Z_91><Z_40><Z_47><Z_25><Z_86><Z_56><Z_96><Z_52><Z_49><Z_83><Z_95><Z_82><Z_63><Z_17><ANSWER>

Question: Isabel had 2 pages of math homework and 4 pages of reading homework. If each page had a certain number of problems on it, she had to complete 30 problems total. How many problems were on each page?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
length of a rectangular plot is 10 mtr more than its breadth . if the cost of fencin g the plot at 26.50 per meter is rs . 5300 , what is the length of the plot in mtr ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
length of a rectangular plot is 10 mtr more than its breadth . if the cost of fencin g the plot at 26.50 per meter is rs . 5300 , what is the length of the plot in mtr ?<|im_end|>
<|im_start|>assistant
<Z_35><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
length of a rectangular plot is 10 mtr more than its breadth . if the cost of fencin g the plot at 26.50 per meter is rs . 5300 , what is the length of the plot in mtr ?<|im_end|>
<|im_start|>assistant
<Z_116><ANSWER>

Question: length of a rectangular plot is 10 mtr more than its breadth . if the cost of fencin g the plot at 26.50 per meter is rs . 5300 , what is the length of the plot in mtr ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the average of three numbers $x\_1$, $x\_2$, and $x\_3$ is $\overline{x} = 40$, find the average of the sample $x\_1 + \overline{x}$, $x\_2 + \overline{x}$, and $x\_3 + \overline{x}$.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the average of three numbers $x\_1$, $x\_2$, and $x\_3$ is $\overline{x} = 40$, find the average of the sample $x\_1 + \overline{x}$, $x\_2 + \overline{x}$, and $x\_3 + \overline{x}$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_65><Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the average of three numbers $x\_1$, $x\_2$, and $x\_3$ is $\overline{x} = 40$, find the average of the sample $x\_1 + \overline{x}$, $x\_2 + \overline{x}$, and $x\_3 + \overline{x}$.<|im_end|>
<|im_start|>assistant
<Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the average of three numbers $x\_1$, $x\_2$, and $x\_3$ is $\overline{x} = 40$, find the average of the sample $x\_1 + \overline{x}$, $x\_2 + \overline{x}$, and $x\_3 + \overline{x}$.<|im_end|>
<|im_start|>assistant
<Z_37><Z_106><Z_3><ANSWER>

Question: Given that the average of three numbers $x\_1$, $x\_2$, and $x\_3$ is $\overline{x} = 40$, find the average of the sample $x\_1 + \overline{x}$, $x\_2 + \overline{x}$, and $x\_3 + \overline{x}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Forty cylinders, each with a diameter of 1 cm and the same height, are tightly packed into a box in 5 rows of 8 cylinders each so that they do not "rattle" during transportation. How many cylinders need to be removed from the box so that, by moving the remaining cylinders and adding the removed cylinders at the end along with one additional cylinder, 41 cylinders of the same size can be packed into the box? Will the cylinders "rattle" in this case?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Forty cylinders, each with a diameter of 1 cm and the same height, are tightly packed into a box in 5 rows of 8 cylinders each so that they do not "rattle" during transportation. How many cylinders need to be removed from the box so that, by moving the remaining cylinders and adding the removed cylinders at the end along with one additional cylinder, 41 cylinders of the same size can be packed into the box? Will the cylinders "rattle" in this case?<|im_end|>
<|im_start|>assistant
<Z_113><Z_105><Z_114><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Forty cylinders, each with a diameter of 1 cm and the same height, are tightly packed into a box in 5 rows of 8 cylinders each so that they do not "rattle" during transportation. How many cylinders need to be removed from the box so that, by moving the remaining cylinders and adding the removed cylinders at the end along with one additional cylinder, 41 cylinders of the same size can be packed into the box? Will the cylinders "rattle" in this case?<|im_end|>
<|im_start|>assistant
<Z_2><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Forty cylinders, each with a diameter of 1 cm and the same height, are tightly packed into a box in 5 rows of 8 cylinders each so that they do not "rattle" during transportation. How many cylinders need to be removed from the box so that, by moving the remaining cylinders and adding the removed cylinders at the end along with one additional cylinder, 41 cylinders of the same size can be packed into the box? Will the cylinders "rattle" in this case?<|im_end|>
<|im_start|>assistant
<Z_30><Z_97><Z_108><Z_28><ANSWER>

Question: Forty cylinders, each with a diameter of 1 cm and the same height, are tightly packed into a box in 5 rows of 8 cylinders each so that they do not "rattle" during transportation. How many cylinders need to be removed from the box so that, by moving the remaining cylinders and adding the removed cylinders at the end along with one additional cylinder, 41 cylinders of the same size can be packed into the box? Will the cylinders "rattle" in this case?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Handsome memorized over a hundred words in seven days. The ratio of the number of words memorized in the first three days to the number of words memorized in the last four days is $3:4$, and the ratio of the number of words memorized in the last three days to the number of words memorized in the first four days is $5:6$. How many words did Handsome memorize on the fourth day?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Handsome memorized over a hundred words in seven days. The ratio of the number of words memorized in the first three days to the number of words memorized in the last four days is $3:4$, and the ratio of the number of words memorized in the last three days to the number of words memorized in the first four days is $5:6$. How many words did Handsome memorize on the fourth day?<|im_end|>
<|im_start|>assistant
<Z_94><Z_97><Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Handsome memorized over a hundred words in seven days. The ratio of the number of words memorized in the first three days to the number of words memorized in the last four days is $3:4$, and the ratio of the number of words memorized in the last three days to the number of words memorized in the first four days is $5:6$. How many words did Handsome memorize on the fourth day?<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Handsome memorized over a hundred words in seven days. The ratio of the number of words memorized in the first three days to the number of words memorized in the last four days is $3:4$, and the ratio of the number of words memorized in the last three days to the number of words memorized in the first four days is $5:6$. How many words did Handsome memorize on the fourth day?<|im_end|>
<|im_start|>assistant
<Z_104><Z_69><Z_13><ANSWER>

Question: Handsome memorized over a hundred words in seven days. The ratio of the number of words memorized in the first three days to the number of words memorized in the last four days is $3:4$, and the ratio of the number of words memorized in the last three days to the number of words memorized in the first four days is $5:6$. How many words did Handsome memorize on the fourth day?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it takes 30 days to fill a laboratory dish with bacteria . if the size of the bacteria doubles each day , how long did it take for the bacteria to fill one sixteenth of the dish ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it takes 30 days to fill a laboratory dish with bacteria . if the size of the bacteria doubles each day , how long did it take for the bacteria to fill one sixteenth of the dish ?<|im_end|>
<|im_start|>assistant
<Z_104><Z_37><Z_73><Z_65><Z_64><Z_59><Z_41><Z_79><Z_66><Z_95><Z_93><Z_91><Z_90><Z_123><Z_72><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it takes 30 days to fill a laboratory dish with bacteria . if the size of the bacteria doubles each day , how long did it take for the bacteria to fill one sixteenth of the dish ?<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
it takes 30 days to fill a laboratory dish with bacteria . if the size of the bacteria doubles each day , how long did it take for the bacteria to fill one sixteenth of the dish ?<|im_end|>
<|im_start|>assistant
<Z_33><Z_31><Z_67><Z_107><Z_27><Z_1><Z_39><Z_49><Z_110><Z_97><Z_27><Z_107><Z_32><Z_61><Z_67><Z_127><ANSWER>

Question: it takes 30 days to fill a laboratory dish with bacteria . if the size of the bacteria doubles each day , how long did it take for the bacteria to fill one sixteenth of the dish ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A big container is 30% full with water. If 54 liters of water is added, the container becomes 3/4 full. What is the capacity of the big container?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A big container is 30% full with water. If 54 liters of water is added, the container becomes 3/4 full. What is the capacity of the big container?<|im_end|>
<|im_start|>assistant
<Z_94><Z_38><Z_89><Z_49><Z_64><Z_68><Z_59><Z_37><Z_90><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A big container is 30% full with water. If 54 liters of water is added, the container becomes 3/4 full. What is the capacity of the big container?<|im_end|>
<|im_start|>assistant
<Z_7><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A big container is 30% full with water. If 54 liters of water is added, the container becomes 3/4 full. What is the capacity of the big container?<|im_end|>
<|im_start|>assistant
<Z_109><Z_107><Z_110><Z_81><Z_84><Z_120><Z_41><Z_46><Z_88><Z_51><ANSWER>

Question: A big container is 30% full with water. If 54 liters of water is added, the container becomes 3/4 full. What is the capacity of the big container?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An engineer undertakes a project to build a road 10 km long in 150 days and employs 30 men for the purpose. After 50 days, he finds only 2 km of the road has been completed. How many extra men must he employ to finish the work in time?<|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
An engineer undertakes a project to build a road 10 km long in 150 days and employs 30 men for the purpose. After 50 days, he finds only 2 km of the road has been completed. How many extra men must he employ to finish the work in time?<|im_end|>
<|im_start|>assistant
<Z_113><Z_89><Z_73><Z_108><Z_43><Z_68><Z_34><Z_37><Z_60><Z_75><Z_55><Z_34><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An engineer undertakes a project to build a road 10 km long in 150 days and employs 30 men for the purpose. After 50 days, he finds only 2 km of the road has been completed. How many extra men must he employ to finish the work in time?<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An engineer undertakes a project to build a road 10 km long in 150 days and employs 30 men for the purpose. After 50 days, he finds only 2 km of the road has been completed. How many extra men must he employ to finish the work in time?<|im_end|>
<|im_start|>assistant
<Z_97><Z_79><Z_72><Z_27><Z_46><Z_65><Z_111><Z_29><Z_22><Z_119><Z_47><Z_24><Z_104><ANSWER>

Question: An engineer undertakes a project to build a road 10 km long in 150 days and employs 30 men for the purpose. After 50 days, he finds only 2 km of the road has been completed. How many extra men must he employ to finish the work in time?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Analogous to arithmetic progression, an "equal sum sequence" is defined as: in a sequence, if the sum of each term and its subsequent term equals the same constant, then this sequence is called an equal sum sequence, and that constant is called the common sum. It is given that the sequence $\{a_n\}$ is an equal sum sequence with $a_1=2$ and the common sum is $5$. Find the sum of the first $2017$ terms, $S_{2017}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Analogous to arithmetic progression, an "equal sum sequence" is defined as: in a sequence, if the sum of each term and its subsequent term equals the same constant, then this sequence is called an equal sum sequence, and that constant is called the common sum. It is given that the sequence $\{a_n\}$ is an equal sum sequence with $a_1=2$ and the common sum is $5$. Find the sum of the first $2017$ terms, $S_{2017}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_113><Z_81><Z_78><Z_70><Z_125><Z_41><Z_118><Z_88><Z_117><Z_76><Z_83><Z_118><Z_66><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Analogous to arithmetic progression, an "equal sum sequence" is defined as: in a sequence, if the sum of each term and its subsequent term equals the same constant, then this sequence is called an equal sum sequence, and that constant is called the common sum. It is given that the sequence $\{a_n\}$ is an equal sum sequence with $a_1=2$ and the common sum is $5$. Find the sum of the first $2017$ terms, $S_{2017}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Analogous to arithmetic progression, an "equal sum sequence" is defined as: in a sequence, if the sum of each term and its subsequent term equals the same constant, then this sequence is called an equal sum sequence, and that constant is called the common sum. It is given that the sequence $\{a_n\}$ is an equal sum sequence with $a_1=2$ and the common sum is $5$. Find the sum of the first $2017$ terms, $S_{2017}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_51><Z_57><Z_30><Z_71><Z_117><Z_71><Z_26><Z_0><Z_7><Z_77><Z_17><Z_22><Z_11><Z_99><ANSWER>

Question: Analogous to arithmetic progression, an "equal sum sequence" is defined as: in a sequence, if the sum of each term and its subsequent term equals the same constant, then this sequence is called an equal sum sequence, and that constant is called the common sum. It is given that the sequence $\{a_n\}$ is an equal sum sequence with $a_1=2$ and the common sum is $5$. Find the sum of the first $2017$ terms, $S_{2017}=$ ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 20 fishermen in the lake who had cast their net to catch fish. They caught 10000 fish in total, and 19 of them caught 400 fish each with their own net. How many fish did the twentieth fisherman catch?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 20 fishermen in the lake who had cast their net to catch fish. They caught 10000 fish in total, and 19 of them caught 400 fish each with their own net. How many fish did the twentieth fisherman catch?<|im_end|>
<|im_start|>assistant
<Z_104><Z_89><Z_120><Z_37><Z_20><Z_39><Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 20 fishermen in the lake who had cast their net to catch fish. They caught 10000 fish in total, and 19 of them caught 400 fish each with their own net. How many fish did the twentieth fisherman catch?<|im_end|>
<|im_start|>assistant
<Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 20 fishermen in the lake who had cast their net to catch fish. They caught 10000 fish in total, and 19 of them caught 400 fish each with their own net. How many fish did the twentieth fisherman catch?<|im_end|>
<|im_start|>assistant
<Z_36><Z_38><Z_81><Z_63><Z_113><Z_69><Z_70><ANSWER>

Question: There were 20 fishermen in the lake who had cast their net to catch fish. They caught 10000 fish in total, and 19 of them caught 400 fish each with their own net. How many fish did the twentieth fisherman catch?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle with circumcenter  $O$  and incenter  $I$ , and suppose that  $OI$  meets  $AB$  and  $AC$  at  $P$  and  $Q$ , respectively. There exists a point  $R$  on arc  $\widehat{BAC}$  such that the circumcircles of triangles  $PQR$  and  $ABC$  are tangent. Given that  $AB = 14$ ,  $BC = 20$ , and  $CA = 26$ , find  $\frac{RC}{RB}$ .

*Proposed by Andrew Wu*<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle with circumcenter  $O$  and incenter  $I$ , and suppose that  $OI$  meets  $AB$  and  $AC$  at  $P$  and  $Q$ , respectively. There exists a point  $R$  on arc  $\widehat{BAC}$  such that the circumcircles of triangles  $PQR$  and  $ABC$  are tangent. Given that  $AB = 14$ ,  $BC = 20$ , and  $CA = 26$ , find  $\frac{RC}{RB}$ .

*Proposed by Andrew Wu*<|im_end|>
<|im_start|>assistant
<Z_94><Z_62><Z_124><Z_114><Z_71><Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle with circumcenter  $O$  and incenter  $I$ , and suppose that  $OI$  meets  $AB$  and  $AC$  at  $P$  and  $Q$ , respectively. There exists a point  $R$  on arc  $\widehat{BAC}$  such that the circumcircles of triangles  $PQR$  and  $ABC$  are tangent. Given that  $AB = 14$ ,  $BC = 20$ , and  $CA = 26$ , find  $\frac{RC}{RB}$ .

*Proposed by Andrew Wu*<|im_end|>
<|im_start|>assistant
<Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ABC$  be a triangle with circumcenter  $O$  and incenter  $I$ , and suppose that  $OI$  meets  $AB$  and  $AC$  at  $P$  and  $Q$ , respectively. There exists a point  $R$  on arc  $\widehat{BAC}$  such that the circumcircles of triangles  $PQR$  and  $ABC$  are tangent. Given that  $AB = 14$ ,  $BC = 20$ , and  $CA = 26$ , find  $\frac{RC}{RB}$ .

*Proposed by Andrew Wu*<|im_end|>
<|im_start|>assistant
<Z_21><Z_46><Z_47><Z_70><Z_119><Z_101><ANSWER>

Question:
Let  $ABC$  be a triangle with circumcenter  $O$  and incenter  $I$ , and suppose that  $OI$  meets  $AB$  and  $AC$  at  $P$  and  $Q$ , respectively. There exists a point  $R$  on arc  $\widehat{BAC}$  such that the circumcircles of triangles  $PQR$  and  $ABC$  are tangent. Given that  $AB = 14$ ,  $BC = 20$ , and  $CA = 26$ , find  $\frac{RC}{RB}$ .

*Proposed by Andrew Wu*


<|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 all the positive divisors of 143?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all the positive divisors of 143?<|im_end|>
<|im_start|>assistant
<Z_57><Z_40><Z_1><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 all the positive divisors of 143?<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all the positive divisors of 143?<|im_end|>
<|im_start|>assistant
<Z_1><Z_92><Z_1><ANSWER>

Question: What is the sum of all the positive divisors of 143?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 6 red peaches, 90 yellow peaches, and some green peaches in the basket. There are 22 red and green peaches in the basket. How many green peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 6 red peaches, 90 yellow peaches, and some green peaches in the basket. There are 22 red and green peaches in the basket. How many green peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_104><Z_107><Z_43><Z_34><Z_33><Z_47><Z_125><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 6 red peaches, 90 yellow peaches, and some green peaches in the basket. There are 22 red and green peaches in the basket. How many green peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 6 red peaches, 90 yellow peaches, and some green peaches in the basket. There are 22 red and green peaches in the basket. How many green peaches are in the basket?<|im_end|>
<|im_start|>assistant
<Z_126><Z_72><Z_56><Z_61><Z_52><Z_30><Z_106><Z_14><ANSWER>

Question: There are 6 red peaches, 90 yellow peaches, and some green peaches in the basket. There are 22 red and green peaches in the basket. How many green peaches are in the basket?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Points \(P\) and \(R\) are located at \( (2, 1) \) and \( (12, 15) \) respectively. Point \(M\) is the midpoint of segment \(\overline{PR}\). Segment \(\overline{PR}\) is reflected over the \(y\)-axis. What is the sum of the coordinates of the image of point \(M\) (the midpoint of the reflected segment)?<|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
Points \(P\) and \(R\) are located at \( (2, 1) \) and \( (12, 15) \) respectively. Point \(M\) is the midpoint of segment \(\overline{PR}\). Segment \(\overline{PR}\) is reflected over the \(y\)-axis. What is the sum of the coordinates of the image of point \(M\) (the midpoint of the reflected segment)?<|im_end|>
<|im_start|>assistant
<Z_104><Z_125><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Points \(P\) and \(R\) are located at \( (2, 1) \) and \( (12, 15) \) respectively. Point \(M\) is the midpoint of segment \(\overline{PR}\). Segment \(\overline{PR}\) is reflected over the \(y\)-axis. What is the sum of the coordinates of the image of point \(M\) (the midpoint of the reflected segment)?<|im_end|>
<|im_start|>assistant
<Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Points \(P\) and \(R\) are located at \( (2, 1) \) and \( (12, 15) \) respectively. Point \(M\) is the midpoint of segment \(\overline{PR}\). Segment \(\overline{PR}\) is reflected over the \(y\)-axis. What is the sum of the coordinates of the image of point \(M\) (the midpoint of the reflected segment)?<|im_end|>
<|im_start|>assistant
<Z_96><Z_39><Z_69><ANSWER>

Question: Points \(P\) and \(R\) are located at \( (2, 1) \) and \( (12, 15) \) respectively. Point \(M\) is the midpoint of segment \(\overline{PR}\). Segment \(\overline{PR}\) is reflected over the \(y\)-axis. What is the sum of the coordinates of the image of point \(M\) (the midpoint of the reflected segment)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 397 butterflies. Each butterfly has 12 black dots and 17 yellow dots. There are _____ black dots in all.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 397 butterflies. Each butterfly has 12 black dots and 17 yellow dots. There are _____ black dots in all.<|im_end|>
<|im_start|>assistant
<Z_94><Z_65><Z_58><Z_52><Z_86><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 397 butterflies. Each butterfly has 12 black dots and 17 yellow dots. There are _____ black dots in all.<|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
There are 397 butterflies. Each butterfly has 12 black dots and 17 yellow dots. There are _____ black dots in all.<|im_end|>
<|im_start|>assistant
<Z_38><Z_82><Z_11><Z_69><Z_49><Z_105><ANSWER>

Question: There are 397 butterflies. Each butterfly has 12 black dots and 17 yellow dots. There are _____ black dots in all.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A supply of modeling clay comes in blocks that are 8 inches by 3 inches by 2 inches. Determine the number of whole blocks needed to create a cylindrical sculpture that is 10 inches high with a diameter of 6 inches.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A supply of modeling clay comes in blocks that are 8 inches by 3 inches by 2 inches. Determine the number of whole blocks needed to create a cylindrical sculpture that is 10 inches high with a diameter of 6 inches.<|im_end|>
<|im_start|>assistant
<Z_113><Z_49><Z_34><Z_107><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A supply of modeling clay comes in blocks that are 8 inches by 3 inches by 2 inches. Determine the number of whole blocks needed to create a cylindrical sculpture that is 10 inches high with a diameter of 6 inches.<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A supply of modeling clay comes in blocks that are 8 inches by 3 inches by 2 inches. Determine the number of whole blocks needed to create a cylindrical sculpture that is 10 inches high with a diameter of 6 inches.<|im_end|>
<|im_start|>assistant
<Z_74><Z_54><Z_11><Z_96><Z_9><ANSWER>

Question: A supply of modeling clay comes in blocks that are 8 inches by 3 inches by 2 inches. Determine the number of whole blocks needed to create a cylindrical sculpture that is 10 inches high with a diameter of 6 inches.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For any real number \( x \), let \( \lceil x \rceil \) denote the smallest integer that is greater than or equal to \( x \) and \( \lfloor x \rfloor \) denote the largest integer that is less than or equal to \( x \) (for example, \( \lceil 1.23 \rceil = 2 \) and \( \lfloor 1.23 \rfloor = 1 \)). Find the value of

\[
\sum_{k=1}^{2010}\left[\frac{2010}{k} - \left\lfloor\frac{2010}{k}\right\rfloor \right].
\]<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For any real number \( x \), let \( \lceil x \rceil \) denote the smallest integer that is greater than or equal to \( x \) and \( \lfloor x \rfloor \) denote the largest integer that is less than or equal to \( x \) (for example, \( \lceil 1.23 \rceil = 2 \) and \( \lfloor 1.23 \rfloor = 1 \)). Find the value of

\[
\sum_{k=1}^{2010}\left[\frac{2010}{k} - \left\lfloor\frac{2010}{k}\right\rfloor \right].
\]<|im_end|>
<|im_start|>assistant
<Z_104><Z_59><Z_79><Z_96><Z_65><Z_120><Z_83><Z_70><Z_82><Z_107><Z_92><Z_41><Z_115><Z_33><Z_117><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For any real number \( x \), let \( \lceil x \rceil \) denote the smallest integer that is greater than or equal to \( x \) and \( \lfloor x \rfloor \) denote the largest integer that is less than or equal to \( x \) (for example, \( \lceil 1.23 \rceil = 2 \) and \( \lfloor 1.23 \rfloor = 1 \)). Find the value of

\[
\sum_{k=1}^{2010}\left[\frac{2010}{k} - \left\lfloor\frac{2010}{k}\right\rfloor \right].
\]<|im_end|>
<|im_start|>assistant
<Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For any real number \( x \), let \( \lceil x \rceil \) denote the smallest integer that is greater than or equal to \( x \) and \( \lfloor x \rfloor \) denote the largest integer that is less than or equal to \( x \) (for example, \( \lceil 1.23 \rceil = 2 \) and \( \lfloor 1.23 \rfloor = 1 \)). Find the value of

\[
\sum_{k=1}^{2010}\left[\frac{2010}{k} - \left\lfloor\frac{2010}{k}\right\rfloor \right].
\]<|im_end|>
<|im_start|>assistant
<Z_71><Z_61><Z_64><Z_3><Z_61><Z_51><Z_72><Z_15><Z_14><Z_69><Z_48><Z_120><Z_6><Z_50><Z_99><ANSWER>

Question:
For any real number \( x \), let \( \lceil x \rceil \) denote the smallest integer that is greater than or equal to \( x \) and \( \lfloor x \rfloor \) denote the largest integer that is less than or equal to \( x \) (for example, \( \lceil 1.23 \rceil = 2 \) and \( \lfloor 1.23 \rfloor = 1 \)). Find the value of

\[
\sum_{k=1}^{2010}\left[\frac{2010}{k} - \left\lfloor\frac{2010}{k}\right\rfloor \right].
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years?<|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
An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years?<|im_end|>
<|im_start|>assistant
<Z_106><Z_66><Z_12><Z_83><Z_81><Z_72><Z_39><Z_39><Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years?<|im_end|>
<|im_start|>assistant
<Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years?<|im_end|>
<|im_start|>assistant
<Z_125><Z_12><Z_105><Z_53><Z_81><Z_113><Z_53><Z_67><Z_3><ANSWER>

Question: An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The white rabbit can hop 15 meters in one minute. The brown rabbit hops a certain distance per minute. The total distance the two rabbits will hop in 5 minutes is 135 meters. How far can the brown rabbit hop in one minute?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The white rabbit can hop 15 meters in one minute. The brown rabbit hops a certain distance per minute. The total distance the two rabbits will hop in 5 minutes is 135 meters. How far can the brown rabbit hop in one minute?<|im_end|>
<|im_start|>assistant
<Z_57><Z_106><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The white rabbit can hop 15 meters in one minute. The brown rabbit hops a certain distance per minute. The total distance the two rabbits will hop in 5 minutes is 135 meters. How far can the brown rabbit hop in one minute?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The white rabbit can hop 15 meters in one minute. The brown rabbit hops a certain distance per minute. The total distance the two rabbits will hop in 5 minutes is 135 meters. How far can the brown rabbit hop in one minute?<|im_end|>
<|im_start|>assistant
<Z_29><Z_34><Z_11><ANSWER>

Question: The white rabbit can hop 15 meters in one minute. The brown rabbit hops a certain distance per minute. The total distance the two rabbits will hop in 5 minutes is 135 meters. How far can the brown rabbit hop in one minute?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 5 students standing in a line. Find the number of arrangements where a certain student does not stand at the head of the line. (Answer with a number).<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 5 students standing in a line. Find the number of arrangements where a certain student does not stand at the head of the line. (Answer with a number).<|im_end|>
<|im_start|>assistant
<Z_94><Z_62><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 5 students standing in a line. Find the number of arrangements where a certain student does not stand at the head of the line. (Answer with a number).<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 5 students standing in a line. Find the number of arrangements where a certain student does not stand at the head of the line. (Answer with a number).<|im_end|>
<|im_start|>assistant
<Z_122><Z_111><ANSWER>

Question: There are 5 students standing in a line. Find the number of arrangements where a certain student does not stand at the head of the line. (Answer with a number).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$<|im_end|>
<|im_start|>assistant
<Z_113><Z_36><Z_119><Z_62><Z_120><Z_126><Z_53><Z_65><Z_93><Z_117><Z_10><Z_41><Z_78><Z_79><Z_66><Z_66><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$<|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 \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$<|im_end|>
<|im_start|>assistant
<Z_126><Z_37><Z_24><Z_20><Z_126><Z_93><Z_69><Z_52><Z_89><Z_13><Z_93><Z_0><Z_101><Z_16><Z_8><Z_83><Z_12><ANSWER>

Question: Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$