<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|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
Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

Question: Ned had 15 video games but some of them weren't working. If he wanted to sell the working games for $7 each and could earn $63, how many video games were not working?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|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
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_104><Z_52><Z_50><Z_91><Z_87><Z_12><Z_8><Z_79><Z_76><Z_63><Z_11><Z_11><Z_38><Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_102><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?<|im_end|>
<|im_start|>assistant
<Z_12><Z_81><Z_3><Z_20><Z_80><Z_69><Z_94><Z_8><Z_121><Z_92><Z_59><Z_34><Z_44><Z_120><ANSWER>

Question: Tom drives from town Q to town B, driving at a certain speed. From town B, Tom immediately continues to town C. The distance between Q and B is twice the distance between B and C. The average speed of the whole journey was 36 mph. Tom's speed driving from B to C is 20 miles per hour. What is Tom's speed driving from Q to B in miles per hour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|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 pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><Z_82><Z_99><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?<|im_end|>
<|im_start|>assistant
<Z_86><Z_103><Z_86><Z_2><Z_2><ANSWER>

Question: A pilot flew an average of 300 mph on a flight out. On the return flight to the same airport, he flew at a certain speed. The total flight time was 8 hours. He flew 1500 miles each way. What was the speed of the pilot on the return flight?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_32><Z_40><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_115><Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.<|im_end|>
<|im_start|>assistant
<Z_124><Z_14><Z_84><Z_54><ANSWER>

Question: The Tri-Star System Mathematics Convention (TSMC) is attended by a committee of five Zetonians, five Pultonians, and five Earthlings. They are sitting around a round table with chairs numbered from $1$ to $15$ in clockwise order. According to convention rules, a Zetonian must occupy chair $1$, a Pultonian must occupy chair $15$, and furthermore, no Earthling can sit immediately to the left of a Zetonian, no Zetonian can sit immediately to the left of a Pultonian, and no Pultonian can sit immediately to the left of an Earthling. Determine the total number of possible seating arrangements for the committee if it can be expressed as $N \cdot (5!)^3$. Provide the value of $N$.


<|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 positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_94><Z_50><Z_110><Z_103><Z_99><Z_62><Z_70><Z_98><Z_63><Z_36><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?<|im_end|>
<|im_start|>assistant
<Z_112><Z_110><Z_103><Z_96><Z_116><Z_55><Z_1><Z_85><Z_14><Z_111><Z_60><ANSWER>

Question: What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=72$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|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 gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_113><Z_1><Z_125><Z_93><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_102><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?<|im_end|>
<|im_start|>assistant
<Z_26><Z_61><Z_42><Z_89><Z_114><ANSWER>

Question: A gumball machine has red, green, and blue gumballs. The machine has half as many blue gumballs as red gumballs. For each blue gumball, the machine has 4 times as many green gumballs. If the machine has 16 red gumballs how many gumballs are in the machine?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|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
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|im_end|>
<|im_start|>assistant
<Z_113><Z_1><Z_40><Z_93><Z_107><Z_120><Z_36><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|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
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.<|im_end|>
<|im_start|>assistant
<Z_23><Z_50><Z_105><Z_112><Z_31><Z_20><Z_65><Z_40><ANSWER>

Question:
Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|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 nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_57><Z_87><Z_75><Z_52><Z_60><Z_82><Z_65><Z_126><Z_9><Z_107><Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.<|im_end|>
<|im_start|>assistant
<Z_79><Z_110><Z_48><Z_7><Z_113><Z_60><Z_77><Z_15><Z_78><Z_90><Z_94><ANSWER>

Question: A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|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
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|im_end|>
<|im_start|>assistant
<Z_21><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

Question:
Find the value of \(3 \cdot 7^{-1} + 5 \cdot 13^{-1} \pmod{63}\).

Express your answer as an integer from 0 to 62, inclusive.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_106><Z_97><Z_78><Z_107><Z_52><Z_48><Z_70><Z_101><Z_96><Z_11><Z_90><Z_20><Z_124><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
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_29><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)
<|im_end|>
<|im_start|>assistant
<Z_4><Z_81><Z_27><Z_34><Z_85><Z_36><Z_41><Z_50><Z_22><Z_71><Z_49><Z_19><Z_34><Z_20><Z_127><ANSWER>

Question:
Let  $p, q, r$  and  $s$  be four prime numbers such that   $$ 5 <p <q <r <s <p + 10. $$
Prove that the sum of the four prime numbers is divisible by  $60$ .

(Walther Janous)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|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
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|im_end|>
<|im_start|>assistant
<Z_104><Z_114><Z_72><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|im_end|>
<|im_start|>assistant
<Z_11><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?<|im_end|>
<|im_start|>assistant
<Z_17><Z_32><Z_54><Z_51><ANSWER>

Question: Kevin starts with 20 cards. He finds 47 cards at the park, but on the way home he loses 7 cards. When he checks his pockets later, he realizes he lost 12 more cards. That night, he wins 15 cards in a game with his friends. How many cards does Kevin have in the end?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|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
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|im_end|>
<|im_start|>assistant
<Z_57><Z_105><Z_87><Z_43><Z_103><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|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
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?<|im_end|>
<|im_start|>assistant
<Z_96><Z_107><Z_111><Z_2><Z_73><Z_54><ANSWER>

Question:
Seven chests are placed in a circle, each containing a certain number of coins. The diagram shows how many coins are in each chest.

In one move, you are allowed to transfer one coin to an adjacent chest. What is the minimum number of moves required to equalize the number of coins in all chests?


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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_104><Z_29><ANSWER>

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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_29><Z_81><Z_99><Z_100><Z_125><Z_36><Z_96><Z_61><Z_11><Z_82><Z_92><Z_125><Z_115><Z_56><Z_20><Z_55><Z_58><Z_120><Z_22><Z_82><ANSWER>

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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_127><Z_2><ANSWER>

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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.<|im_end|>
<|im_start|>assistant
<Z_0><Z_11><Z_86><Z_9><Z_14><Z_36><Z_88><Z_117><Z_56><Z_5><Z_78><Z_46><Z_60><Z_68><Z_16><Z_38><Z_66><Z_110><Z_64><Z_110><ANSWER>

Question:
For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|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
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|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
Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?<|im_end|>
<|im_start|>assistant
<Z_92><ANSWER>

Question: Trevor is a comic book illustrator. In the past three months, he has drawn some pages of the comic. The comic he illustrates releases a new issue once a month. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. If the first issue of the comic was 72 pages, how many pages did Trevor draw in the past three months?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_57><Z_87><Z_70><Z_70><Z_64><Z_73><Z_32><Z_91><Z_9><Z_47><Z_32><Z_124><Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.<|im_end|>
<|im_start|>assistant
<Z_56><Z_78><Z_80><Z_86><Z_17><Z_57><Z_38><Z_33><Z_118><Z_83><Z_71><Z_78><Z_22><ANSWER>

Question: Among $a$, $b$, $c$, $d$, $e$—a total of $5$ people—select $1$ as the team leader and $1$ as the deputy team leader, but $a$ cannot be the deputy team leader. The total number of different selection methods is ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|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 rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|im_end|>
<|im_start|>assistant
<Z_104><Z_13><Z_73><Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|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
A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.<|im_end|>
<|im_start|>assistant
<Z_41><Z_25><Z_21><Z_84><ANSWER>

Question: A rectangle has a perimeter of 28 centimeters (cm) and a width of 6 centimeters (cm). Find the area of this rectangle.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventeen is the seventh prime number. What is the twentieth prime 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
Seventeen is the seventh prime number. What is the twentieth prime number?<|im_end|>
<|im_start|>assistant
<Z_94><Z_93><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventeen is the seventh prime number. What is the twentieth prime number?<|im_end|>
<|im_start|>assistant
<Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventeen is the seventh prime number. What is the twentieth prime number?<|im_end|>
<|im_start|>assistant
<Z_110><Z_3><Z_60><ANSWER>

Question: Seventeen is the seventh prime number. What is the twentieth prime number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|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
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_57><Z_108><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?<|im_end|>
<|im_start|>assistant
<Z_48><Z_34><Z_53><ANSWER>

Question: Adam could fit 11 action figures on each shelf in his room. His room has a certain number of shelves and 40 cabinets. His shelves could hold a total of 44 action figures. How many shelves does his room have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_104><Z_38><Z_1><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|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
In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.<|im_end|>
<|im_start|>assistant
<Z_80><Z_96><Z_71><ANSWER>

Question: In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|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 was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_113><Z_78><Z_36><Z_119><Z_34><Z_41><Z_95><Z_26><Z_26><Z_78><Z_107><Z_117><Z_65><Z_79><Z_56><Z_60><Z_91><Z_19><Z_126><Z_79><Z_78><Z_48><Z_70><Z_119><Z_12><Z_28><Z_96><Z_78><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.<|im_end|>
<|im_start|>assistant
<Z_126><Z_25><Z_4><Z_116><Z_41><Z_38><Z_106><Z_62><Z_32><Z_11><Z_91><Z_35><Z_6><Z_38><Z_33><Z_106><Z_98><Z_21><Z_114><Z_6><Z_104><Z_6><Z_31><Z_65><Z_6><Z_17><Z_17><Z_67><Z_114><ANSWER>

Question: There was a mistake in the calculation, multiplying the number by 5 and then adding 7 instead of multiplying the sum of the number and 5 by 7. Then, the wrong result came out as 27. Find the correct calculated value.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|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
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|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
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|im_end|>
<|im_start|>assistant
<Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

Question: Malar and Roja combined will complete a task in some days, but Malar alone can complete the same work in 60 days. Roja can complete this work in 84 days. In how many days can Malar and Roja complete the task together?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_94><Z_106><Z_33><Z_12><Z_89><Z_64><Z_12><Z_32><Z_53><Z_125><Z_70><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  <|im_end|>
<|im_start|>assistant
<Z_105><Z_10><Z_38><Z_65><Z_23><Z_51><Z_34><Z_90><Z_70><Z_92><Z_120><ANSWER>

Question: Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even divisors of $8!$ are also divisible by $3$?<|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
How many even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_57><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 even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_117><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even divisors of $8!$ are also divisible by $3$?<|im_end|>
<|im_start|>assistant
<Z_109><Z_30><ANSWER>

Question: How many even divisors of $8!$ are also divisible by $3$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The decimal representation of a fraction ends in $0.\overline{6}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction?<|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 decimal representation of a fraction ends in $0.\overline{6}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction?<|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 decimal representation of a fraction ends in $0.\overline{6}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction?<|im_end|>
<|im_start|>assistant
<Z_13><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The decimal representation of a fraction ends in $0.\overline{6}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction?<|im_end|>
<|im_start|>assistant
<Z_64><ANSWER>

Question: The decimal representation of a fraction ends in $0.\overline{6}$. When the decimal is changed to a common fraction and reduced to lowest terms, what is the denominator of the fraction?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Using Mirtes' idea, find the minimum number of employees the sector would need to ensure that two people have the same zodiac sign.

b) What is the minimum number of employees the sector would need to ensure that at least four of them have their birthdays on the same day of the week this year?<|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) Using Mirtes' idea, find the minimum number of employees the sector would need to ensure that two people have the same zodiac sign.

b) What is the minimum number of employees the sector would need to ensure that at least four of them have their birthdays on the same day of the week this year?<|im_end|>
<|im_start|>assistant
<Z_57><Z_105><Z_125><Z_38><Z_40><Z_10><Z_45><Z_91><Z_95><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Using Mirtes' idea, find the minimum number of employees the sector would need to ensure that two people have the same zodiac sign.

b) What is the minimum number of employees the sector would need to ensure that at least four of them have their birthdays on the same day of the week this year?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Using Mirtes' idea, find the minimum number of employees the sector would need to ensure that two people have the same zodiac sign.

b) What is the minimum number of employees the sector would need to ensure that at least four of them have their birthdays on the same day of the week this year?<|im_end|>
<|im_start|>assistant
<Z_48><Z_26><Z_7><Z_99><Z_9><Z_49><Z_17><Z_111><Z_30><Z_91><ANSWER>

Question:
a) Using Mirtes' idea, find the minimum number of employees the sector would need to ensure that two people have the same zodiac sign.

b) What is the minimum number of employees the sector would need to ensure that at least four of them have their birthdays on the same day of the week this year?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five sailors landed on an island and by evening gathered a pile of coconuts. They postponed the division of the coconuts until the morning. One of them, waking up at night, gave one coconut to a monkey, then took exactly one-fifth of the remaining coconuts for himself, then went back to sleep. Each of the other sailors did the same thing during the night, unaware of the actions of the others. In the morning, they divided the remaining coconuts equally, and this time no extra coconut was left for the monkey. What could be the smallest number of coconuts in the collected pile?<|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
Five sailors landed on an island and by evening gathered a pile of coconuts. They postponed the division of the coconuts until the morning. One of them, waking up at night, gave one coconut to a monkey, then took exactly one-fifth of the remaining coconuts for himself, then went back to sleep. Each of the other sailors did the same thing during the night, unaware of the actions of the others. In the morning, they divided the remaining coconuts equally, and this time no extra coconut was left for the monkey. What could be the smallest number of coconuts in the collected pile?<|im_end|>
<|im_start|>assistant
<Z_113><Z_58><Z_11><Z_103><Z_125><Z_114><Z_78><Z_88><Z_76><Z_117><Z_82><Z_91><Z_31><Z_83><Z_103><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five sailors landed on an island and by evening gathered a pile of coconuts. They postponed the division of the coconuts until the morning. One of them, waking up at night, gave one coconut to a monkey, then took exactly one-fifth of the remaining coconuts for himself, then went back to sleep. Each of the other sailors did the same thing during the night, unaware of the actions of the others. In the morning, they divided the remaining coconuts equally, and this time no extra coconut was left for the monkey. What could be the smallest number of coconuts in the collected pile?<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five sailors landed on an island and by evening gathered a pile of coconuts. They postponed the division of the coconuts until the morning. One of them, waking up at night, gave one coconut to a monkey, then took exactly one-fifth of the remaining coconuts for himself, then went back to sleep. Each of the other sailors did the same thing during the night, unaware of the actions of the others. In the morning, they divided the remaining coconuts equally, and this time no extra coconut was left for the monkey. What could be the smallest number of coconuts in the collected pile?<|im_end|>
<|im_start|>assistant
<Z_120><Z_3><Z_66><Z_98><Z_104><Z_79><Z_86><Z_0><Z_97><Z_26><Z_22><Z_95><Z_13><Z_105><Z_37><Z_8><ANSWER>

Question: Five sailors landed on an island and by evening gathered a pile of coconuts. They postponed the division of the coconuts until the morning. One of them, waking up at night, gave one coconut to a monkey, then took exactly one-fifth of the remaining coconuts for himself, then went back to sleep. Each of the other sailors did the same thing during the night, unaware of the actions of the others. In the morning, they divided the remaining coconuts equally, and this time no extra coconut was left for the monkey. What could be the smallest number of coconuts in the collected pile?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiao Ming is solving the problem: given $a=\frac{1}{2+\sqrt{3}}$, find the value of $2a^{2}-8a+1$. He analyzed and answered it as follows:
Since $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$,
Therefore, $a-2=-\sqrt{3}$.
Therefore, $\left(a-2\right)^{2}=3$, which means $a^{2}-4a+4=3$.
Therefore, $a^{2}-4a=-1$,
Therefore, $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$.
According to Xiao Ming's analysis process, please solve the following problems:
$(1)$ Calculate: $\frac{1}{\sqrt{2}+1}=$______；
$(2)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+…+\frac{1}{\sqrt{2024}+\sqrt{2023}}$；
$(3)$ If $a=\frac{1}{\sqrt{5}-2}$, find the value of $2a^{2}-8a+1$.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiao Ming is solving the problem: given $a=\frac{1}{2+\sqrt{3}}$, find the value of $2a^{2}-8a+1$. He analyzed and answered it as follows:
Since $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$,
Therefore, $a-2=-\sqrt{3}$.
Therefore, $\left(a-2\right)^{2}=3$, which means $a^{2}-4a+4=3$.
Therefore, $a^{2}-4a=-1$,
Therefore, $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$.
According to Xiao Ming's analysis process, please solve the following problems:
$(1)$ Calculate: $\frac{1}{\sqrt{2}+1}=$______；
$(2)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+…+\frac{1}{\sqrt{2024}+\sqrt{2023}}$；
$(3)$ If $a=\frac{1}{\sqrt{5}-2}$, find the value of $2a^{2}-8a+1$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiao Ming is solving the problem: given $a=\frac{1}{2+\sqrt{3}}$, find the value of $2a^{2}-8a+1$. He analyzed and answered it as follows:
Since $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$,
Therefore, $a-2=-\sqrt{3}$.
Therefore, $\left(a-2\right)^{2}=3$, which means $a^{2}-4a+4=3$.
Therefore, $a^{2}-4a=-1$,
Therefore, $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$.
According to Xiao Ming's analysis process, please solve the following problems:
$(1)$ Calculate: $\frac{1}{\sqrt{2}+1}=$______；
$(2)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+…+\frac{1}{\sqrt{2024}+\sqrt{2023}}$；
$(3)$ If $a=\frac{1}{\sqrt{5}-2}$, find the value of $2a^{2}-8a+1$.<|im_end|>
<|im_start|>assistant
<Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiao Ming is solving the problem: given $a=\frac{1}{2+\sqrt{3}}$, find the value of $2a^{2}-8a+1$. He analyzed and answered it as follows:
Since $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$,
Therefore, $a-2=-\sqrt{3}$.
Therefore, $\left(a-2\right)^{2}=3$, which means $a^{2}-4a+4=3$.
Therefore, $a^{2}-4a=-1$,
Therefore, $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$.
According to Xiao Ming's analysis process, please solve the following problems:
$(1)$ Calculate: $\frac{1}{\sqrt{2}+1}=$______；
$(2)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+…+\frac{1}{\sqrt{2024}+\sqrt{2023}}$；
$(3)$ If $a=\frac{1}{\sqrt{5}-2}$, find the value of $2a^{2}-8a+1$.<|im_end|>
<|im_start|>assistant
<Z_89><Z_92><ANSWER>

Question:
Xiao Ming is solving the problem: given $a=\frac{1}{2+\sqrt{3}}$, find the value of $2a^{2}-8a+1$. He analyzed and answered it as follows:
Since $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$,
Therefore, $a-2=-\sqrt{3}$.
Therefore, $\left(a-2\right)^{2}=3$, which means $a^{2}-4a+4=3$.
Therefore, $a^{2}-4a=-1$,
Therefore, $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$.
According to Xiao Ming's analysis process, please solve the following problems:
$(1)$ Calculate: $\frac{1}{\sqrt{2}+1}=$______；
$(2)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+…+\frac{1}{\sqrt{2024}+\sqrt{2023}}$；
$(3)$ If $a=\frac{1}{\sqrt{5}-2}$, find the value of $2a^{2}-8a+1$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( a \) is the smallest positive integer which gives a remainder of 1 when divided by 3 and is a multiple of 5, find the value of \( a \).<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( a \) is the smallest positive integer which gives a remainder of 1 when divided by 3 and is a multiple of 5, find the value of \( a \).<|im_end|>
<|im_start|>assistant
<Z_94><Z_13><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( a \) is the smallest positive integer which gives a remainder of 1 when divided by 3 and is a multiple of 5, find the value of \( a \).<|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
If \( a \) is the smallest positive integer which gives a remainder of 1 when divided by 3 and is a multiple of 5, find the value of \( a \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_78><Z_110><ANSWER>

Question: If \( a \) is the smallest positive integer which gives a remainder of 1 when divided by 3 and is a multiple of 5, find the value of \( a \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice is making a big stack of dishes to see how tall she can make it before it topples over and falls. She starts off by placing some plates on top of each other. The tower holds fine so she adds 37 more plates. She is in the process of placing more and more plates when it finally topples over and crashes down, all 83 plates. How many plates did Alice start with?<|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
Alice is making a big stack of dishes to see how tall she can make it before it topples over and falls. She starts off by placing some plates on top of each other. The tower holds fine so she adds 37 more plates. She is in the process of placing more and more plates when it finally topples over and crashes down, all 83 plates. How many plates did Alice start with?<|im_end|>
<|im_start|>assistant
<Z_113><Z_119><Z_100><Z_51><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice is making a big stack of dishes to see how tall she can make it before it topples over and falls. She starts off by placing some plates on top of each other. The tower holds fine so she adds 37 more plates. She is in the process of placing more and more plates when it finally topples over and crashes down, all 83 plates. How many plates did Alice start with?<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice is making a big stack of dishes to see how tall she can make it before it topples over and falls. She starts off by placing some plates on top of each other. The tower holds fine so she adds 37 more plates. She is in the process of placing more and more plates when it finally topples over and crashes down, all 83 plates. How many plates did Alice start with?<|im_end|>
<|im_start|>assistant
<Z_104><Z_37><Z_49><Z_125><Z_125><ANSWER>

Question: Alice is making a big stack of dishes to see how tall she can make it before it topples over and falls. She starts off by placing some plates on top of each other. The tower holds fine so she adds 37 more plates. She is in the process of placing more and more plates when it finally topples over and crashes down, all 83 plates. How many plates did Alice start with?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the arithmetic sequence \(a_{1}, a_{2}, \cdots, a_{n}\), it is known that
\[ \sum_{i=1}^{n}\left|a_{i}\right| = \sum_{i=1}^{n}\left|a_{i}-1\right| = \sum_{i=1}^{n}\left|a_{i}+1\right| = 2021. \]

Find the maximum value of \(n\).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the arithmetic sequence \(a_{1}, a_{2}, \cdots, a_{n}\), it is known that
\[ \sum_{i=1}^{n}\left|a_{i}\right| = \sum_{i=1}^{n}\left|a_{i}-1\right| = \sum_{i=1}^{n}\left|a_{i}+1\right| = 2021. \]

Find the maximum value of \(n\).<|im_end|>
<|im_start|>assistant
<Z_57><Z_40><Z_78><Z_99><Z_33><Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the arithmetic sequence \(a_{1}, a_{2}, \cdots, a_{n}\), it is known that
\[ \sum_{i=1}^{n}\left|a_{i}\right| = \sum_{i=1}^{n}\left|a_{i}-1\right| = \sum_{i=1}^{n}\left|a_{i}+1\right| = 2021. \]

Find the maximum value of \(n\).<|im_end|>
<|im_start|>assistant
<Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the arithmetic sequence \(a_{1}, a_{2}, \cdots, a_{n}\), it is known that
\[ \sum_{i=1}^{n}\left|a_{i}\right| = \sum_{i=1}^{n}\left|a_{i}-1\right| = \sum_{i=1}^{n}\left|a_{i}+1\right| = 2021. \]

Find the maximum value of \(n\).<|im_end|>
<|im_start|>assistant
<Z_60><Z_50><Z_64><Z_9><Z_118><Z_63><ANSWER>

Question:
Given the arithmetic sequence \(a_{1}, a_{2}, \cdots, a_{n}\), it is known that
\[ \sum_{i=1}^{n}\left|a_{i}\right| = \sum_{i=1}^{n}\left|a_{i}-1\right| = \sum_{i=1}^{n}\left|a_{i}+1\right| = 2021. \]

Find the maximum value of \(n\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?<|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
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?<|im_end|>
<|im_start|>assistant
<Z_104><Z_82><Z_80><Z_38><Z_20><Z_87><Z_16><Z_98><Z_76><Z_22><Z_76><Z_78><Z_101><Z_49><Z_101><Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?<|im_end|>
<|im_start|>assistant
<Z_78><Z_47><Z_98><Z_83><Z_93><Z_126><Z_4><Z_96><Z_68><Z_115><Z_61><Z_119><Z_26><Z_65><Z_16><Z_10><ANSWER>

Question: Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mr. Fat needs 25 minutes to eat a pound of cereal, Mr. Thin needs 35 minutes, and a new person, Mr. Medium, needs 28 minutes. If all three eat together, how long will it take them to finish 5 pounds of cereal?<|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
Mr. Fat needs 25 minutes to eat a pound of cereal, Mr. Thin needs 35 minutes, and a new person, Mr. Medium, needs 28 minutes. If all three eat together, how long will it take them to finish 5 pounds of cereal?<|im_end|>
<|im_start|>assistant
<Z_104><Z_108><Z_32><Z_105><Z_76><Z_93><Z_12><Z_49><Z_37><Z_91><Z_32><Z_91><Z_60><Z_107><Z_117><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mr. Fat needs 25 minutes to eat a pound of cereal, Mr. Thin needs 35 minutes, and a new person, Mr. Medium, needs 28 minutes. If all three eat together, how long will it take them to finish 5 pounds of cereal?<|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
Mr. Fat needs 25 minutes to eat a pound of cereal, Mr. Thin needs 35 minutes, and a new person, Mr. Medium, needs 28 minutes. If all three eat together, how long will it take them to finish 5 pounds of cereal?<|im_end|>
<|im_start|>assistant
<Z_49><Z_84><Z_83><Z_28><Z_120><Z_65><Z_48><Z_96><Z_125><Z_91><Z_69><Z_104><Z_30><Z_59><Z_31><Z_23><ANSWER>

Question: Mr. Fat needs 25 minutes to eat a pound of cereal, Mr. Thin needs 35 minutes, and a new person, Mr. Medium, needs 28 minutes. If all three eat together, how long will it take them to finish 5 pounds of cereal?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A caterer ordered some ice-cream bars and 125 sundaes. The total price was $200.00 and the price of each ice-cream bar was $0.60. The price of each sundae was $0.52. How many ice-cream bars did the caterer order?<|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 caterer ordered some ice-cream bars and 125 sundaes. The total price was $200.00 and the price of each ice-cream bar was $0.60. The price of each sundae was $0.52. How many ice-cream bars did the caterer order?<|im_end|>
<|im_start|>assistant
<Z_113><Z_74><Z_74><Z_36><Z_65><Z_96><Z_123><Z_39><Z_55><Z_103><Z_61><Z_76><Z_98><Z_101><Z_10><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A caterer ordered some ice-cream bars and 125 sundaes. The total price was $200.00 and the price of each ice-cream bar was $0.60. The price of each sundae was $0.52. How many ice-cream bars did the caterer order?<|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
A caterer ordered some ice-cream bars and 125 sundaes. The total price was $200.00 and the price of each ice-cream bar was $0.60. The price of each sundae was $0.52. How many ice-cream bars did the caterer order?<|im_end|>
<|im_start|>assistant
<Z_62><Z_56><Z_69><Z_82><Z_98><Z_101><Z_41><Z_40><Z_7><Z_62><Z_113><Z_3><Z_23><Z_26><Z_18><Z_126><ANSWER>

Question: A caterer ordered some ice-cream bars and 125 sundaes. The total price was $200.00 and the price of each ice-cream bar was $0.60. The price of each sundae was $0.52. How many ice-cream bars did the caterer order?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that  $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal  $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?<|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
Suppose that  $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal  $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?<|im_end|>
<|im_start|>assistant
<Z_113><Z_47><Z_99><Z_120><Z_92><Z_37><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that  $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal  $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?<|im_end|>
<|im_start|>assistant
<Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that  $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal  $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?<|im_end|>
<|im_start|>assistant
<Z_20><Z_52><Z_30><Z_14><Z_67><Z_89><Z_44><ANSWER>

Question: Suppose that  $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal  $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the prime factors of 57?<|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 factors of 57?<|im_end|>
<|im_start|>assistant
<Z_104><Z_78><Z_37><Z_50><Z_82><Z_47><Z_123><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 factors of 57?<|im_end|>
<|im_start|>assistant
<Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the prime factors of 57?<|im_end|>
<|im_start|>assistant
<Z_120><Z_23><Z_113><Z_111><Z_90><Z_109><Z_25><ANSWER>

Question: What is the sum of the prime factors of 57?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?<|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 sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?<|im_end|>
<|im_start|>assistant
<Z_94><Z_64><Z_37><Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?<|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?<|im_end|>
<|im_start|>assistant
<Z_124><Z_24><Z_6><Z_52><ANSWER>

Question: The sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, some did not opt for math. 15 did not opt for science and 2 did not opt for either. There are 7 students who opted for both math and science. How many students did not opt for math?<|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 class of 40 students, some did not opt for math. 15 did not opt for science and 2 did not opt for either. There are 7 students who opted for both math and science. How many students did not opt for math?<|im_end|>
<|im_start|>assistant
<Z_57><Z_29><Z_37><Z_100><Z_37><Z_65><Z_88><Z_20><Z_65><Z_124><Z_90><Z_12><Z_10><Z_95><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, some did not opt for math. 15 did not opt for science and 2 did not opt for either. There are 7 students who opted for both math and science. How many students did not opt for math?<|im_end|>
<|im_start|>assistant
<Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, some did not opt for math. 15 did not opt for science and 2 did not opt for either. There are 7 students who opted for both math and science. How many students did not opt for math?<|im_end|>
<|im_start|>assistant
<Z_71><Z_31><Z_20><Z_88><Z_104><Z_36><Z_79><Z_92><Z_64><Z_27><Z_69><Z_125><Z_94><Z_110><Z_103><ANSWER>

Question: In a class of 40 students, some did not opt for math. 15 did not opt for science and 2 did not opt for either. There are 7 students who opted for both math and science. How many students did not opt for math?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Hoseok added 36 to a certain number and got 71. What is the value of adding 10 to this certain number?<|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
Hoseok added 36 to a certain number and got 71. What is the value of adding 10 to this certain number?<|im_end|>
<|im_start|>assistant
<Z_57><Z_105><Z_70><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Hoseok added 36 to a certain number and got 71. What is the value of adding 10 to this certain number?<|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
Hoseok added 36 to a certain number and got 71. What is the value of adding 10 to this certain number?<|im_end|>
<|im_start|>assistant
<Z_121><Z_113><Z_54><ANSWER>

Question: Hoseok added 36 to a certain number and got 71. What is the value of adding 10 to this certain number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a set of weights, all of which have different masses, it is known that if any pair of weights is placed on the left pan, the balance can be achieved by placing one or more of the remaining weights on the right pan. Find the smallest possible number of weights in the set.<|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 set of weights, all of which have different masses, it is known that if any pair of weights is placed on the left pan, the balance can be achieved by placing one or more of the remaining weights on the right pan. Find the smallest possible number of weights in the set.<|im_end|>
<|im_start|>assistant
<Z_57><Z_66><Z_65><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a set of weights, all of which have different masses, it is known that if any pair of weights is placed on the left pan, the balance can be achieved by placing one or more of the remaining weights on the right pan. Find the smallest possible number of weights in the set.<|im_end|>
<|im_start|>assistant
<Z_21><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a set of weights, all of which have different masses, it is known that if any pair of weights is placed on the left pan, the balance can be achieved by placing one or more of the remaining weights on the right pan. Find the smallest possible number of weights in the set.<|im_end|>
<|im_start|>assistant
<Z_117><Z_38><Z_67><Z_30><ANSWER>

Question: In a set of weights, all of which have different masses, it is known that if any pair of weights is placed on the left pan, the balance can be achieved by placing one or more of the remaining weights on the right pan. Find the smallest possible number of weights in the set.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. If all of them started together on the 1st day, and there are 3 women in the family, how many days do they take to complete the work if men work on alternate days and women work every third day?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. If all of them started together on the 1st day, and there are 3 women in the family, how many days do they take to complete the work if men work on alternate days and women work every third day?<|im_end|>
<|im_start|>assistant
<Z_94><Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. If all of them started together on the 1st day, and there are 3 women in the family, how many days do they take to complete the work if men work on alternate days and women work every third day?<|im_end|>
<|im_start|>assistant
<Z_31><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. If all of them started together on the 1st day, and there are 3 women in the family, how many days do they take to complete the work if men work on alternate days and women work every third day?<|im_end|>
<|im_start|>assistant
<Z_124><Z_36><ANSWER>

Question: There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. If all of them started together on the 1st day, and there are 3 women in the family, how many days do they take to complete the work if men work on alternate days and women work every third day?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a hockey league, teams get 2 points for a win, 1 point for a tie, and zero points for a loss. The first-place team has 12 wins and 4 ties, the second-place team has 13 wins and 1 tie, and Elsa's team has some wins and 10 ties. The average number of points for the playoff teams is 27. How many wins does Elsa's team have?<|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 hockey league, teams get 2 points for a win, 1 point for a tie, and zero points for a loss. The first-place team has 12 wins and 4 ties, the second-place team has 13 wins and 1 tie, and Elsa's team has some wins and 10 ties. The average number of points for the playoff teams is 27. How many wins does Elsa's team have?<|im_end|>
<|im_start|>assistant
<Z_94><Z_27><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a hockey league, teams get 2 points for a win, 1 point for a tie, and zero points for a loss. The first-place team has 12 wins and 4 ties, the second-place team has 13 wins and 1 tie, and Elsa's team has some wins and 10 ties. The average number of points for the playoff teams is 27. How many wins does Elsa's team have?<|im_end|>
<|im_start|>assistant
<Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a hockey league, teams get 2 points for a win, 1 point for a tie, and zero points for a loss. The first-place team has 12 wins and 4 ties, the second-place team has 13 wins and 1 tie, and Elsa's team has some wins and 10 ties. The average number of points for the playoff teams is 27. How many wins does Elsa's team have?<|im_end|>
<|im_start|>assistant
<Z_84><Z_94><Z_116><ANSWER>

Question: In a hockey league, teams get 2 points for a win, 1 point for a tie, and zero points for a loss. The first-place team has 12 wins and 4 ties, the second-place team has 13 wins and 1 tie, and Elsa's team has some wins and 10 ties. The average number of points for the playoff teams is 27. How many wins does Elsa's team have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The line $y = -x + 2$ intersects the circle $x^2 + y^2 = 3$ at points A and B. What is the length of segment AB?<|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 line $y = -x + 2$ intersects the circle $x^2 + y^2 = 3$ at points A and B. What is the length of segment AB?<|im_end|>
<|im_start|>assistant
<Z_104><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The line $y = -x + 2$ intersects the circle $x^2 + y^2 = 3$ at points A and B. What is the length of segment AB?<|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 line $y = -x + 2$ intersects the circle $x^2 + y^2 = 3$ at points A and B. What is the length of segment AB?<|im_end|>
<|im_start|>assistant
<Z_90><Z_30><ANSWER>

Question: The line $y = -x + 2$ intersects the circle $x^2 + y^2 = 3$ at points A and B. What is the length of segment AB?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What will be the cost of building a fence around an irregular rectangular plot with dimensions 17 ft by 21 ft, if the price per foot for the fence varies for each side? The prices per foot are as follows: Rs. 59 for the 17 ft sides and Rs. 81 for the 21 ft 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
What will be the cost of building a fence around an irregular rectangular plot with dimensions 17 ft by 21 ft, if the price per foot for the fence varies for each side? The prices per foot are as follows: Rs. 59 for the 17 ft sides and Rs. 81 for the 21 ft sides.<|im_end|>
<|im_start|>assistant
<Z_104><Z_49><Z_66><Z_49><Z_119><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What will be the cost of building a fence around an irregular rectangular plot with dimensions 17 ft by 21 ft, if the price per foot for the fence varies for each side? The prices per foot are as follows: Rs. 59 for the 17 ft sides and Rs. 81 for the 21 ft sides.<|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
What will be the cost of building a fence around an irregular rectangular plot with dimensions 17 ft by 21 ft, if the price per foot for the fence varies for each side? The prices per foot are as follows: Rs. 59 for the 17 ft sides and Rs. 81 for the 21 ft sides.<|im_end|>
<|im_start|>assistant
<Z_85><Z_46><Z_12><Z_36><Z_12><Z_52><ANSWER>

Question: What will be the cost of building a fence around an irregular rectangular plot with dimensions 17 ft by 21 ft, if the price per foot for the fence varies for each side? The prices per foot are as follows: Rs. 59 for the 17 ft sides and Rs. 81 for the 21 ft sides.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the short diagonal of a rhombus is 6 centimeters (cm) and the long diagonal is 10 centimeters (cm), what is the area of this rhombus?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the short diagonal of a rhombus is 6 centimeters (cm) and the long diagonal is 10 centimeters (cm), what is the area of this rhombus?<|im_end|>
<|im_start|>assistant
<Z_104><Z_38><Z_59><Z_99><Z_53><Z_33><Z_37><Z_20><Z_33><Z_34><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the short diagonal of a rhombus is 6 centimeters (cm) and the long diagonal is 10 centimeters (cm), what is the area of this rhombus?<|im_end|>
<|im_start|>assistant
<Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the short diagonal of a rhombus is 6 centimeters (cm) and the long diagonal is 10 centimeters (cm), what is the area of this rhombus?<|im_end|>
<|im_start|>assistant
<Z_49><Z_45><Z_101><Z_70><Z_46><Z_125><Z_20><Z_46><Z_4><Z_113><Z_87><ANSWER>

Question: If the short diagonal of a rhombus is 6 centimeters (cm) and the long diagonal is 10 centimeters (cm), what is the area of this rhombus?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on the independence day , bananas were be equally distributed among the children in a school so that each child would get two bananas . on the particular day 420 children were absent and as a result each child got two extra bananas . find the actual number of children in the school ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on the independence day , bananas were be equally distributed among the children in a school so that each child would get two bananas . on the particular day 420 children were absent and as a result each child got two extra bananas . find the actual number of children in the school ?<|im_end|>
<|im_start|>assistant
<Z_94><Z_37><Z_70><Z_70><Z_99><Z_71><Z_33><Z_114><Z_99><Z_90><Z_118><Z_54><Z_114><Z_22><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on the independence day , bananas were be equally distributed among the children in a school so that each child would get two bananas . on the particular day 420 children were absent and as a result each child got two extra bananas . find the actual number of children in the school ?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
on the independence day , bananas were be equally distributed among the children in a school so that each child would get two bananas . on the particular day 420 children were absent and as a result each child got two extra bananas . find the actual number of children in the school ?<|im_end|>
<|im_start|>assistant
<Z_71><Z_18><Z_50><Z_54><Z_83><Z_14><Z_89><Z_48><Z_78><Z_36><Z_97><Z_56><Z_18><Z_27><Z_10><ANSWER>

Question: on the independence day , bananas were be equally distributed among the children in a school so that each child would get two bananas . on the particular day 420 children were absent and as a result each child got two extra bananas . find the actual number of children in the school ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stacy and Heather are some miles apart and walk towards each other along the same route. Stacy walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. Heather starts her journey 24 minutes after Stacy. How far apart were Stacy and Heather originally if Heather has walked 1.1818181818181817 miles when the two meet?<|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
Stacy and Heather are some miles apart and walk towards each other along the same route. Stacy walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. Heather starts her journey 24 minutes after Stacy. How far apart were Stacy and Heather originally if Heather has walked 1.1818181818181817 miles when the two meet?<|im_end|>
<|im_start|>assistant
<Z_57><Z_78><Z_107><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stacy and Heather are some miles apart and walk towards each other along the same route. Stacy walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. Heather starts her journey 24 minutes after Stacy. How far apart were Stacy and Heather originally if Heather has walked 1.1818181818181817 miles when the two meet?<|im_end|>
<|im_start|>assistant
<Z_74><Z_11><Z_56><Z_93><ANSWER>

Question: Stacy and Heather are some miles apart and walk towards each other along the same route. Stacy walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. Heather starts her journey 24 minutes after Stacy. How far apart were Stacy and Heather originally if Heather has walked 1.1818181818181817 miles when the two meet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If Pam is currently twice as young as Rena is, and in 12 years, Rena will be 7 years older than Jess, who will be 3 times as old as Pam is now. Meanwhile, Matt will be 9 years older than Jess in 8 years and twice as old as Rena in 4 years. How old are Pam, Rena, Jess, and Matt currently?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If Pam is currently twice as young as Rena is, and in 12 years, Rena will be 7 years older than Jess, who will be 3 times as old as Pam is now. Meanwhile, Matt will be 9 years older than Jess in 8 years and twice as old as Rena in 4 years. How old are Pam, Rena, Jess, and Matt currently?<|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
If Pam is currently twice as young as Rena is, and in 12 years, Rena will be 7 years older than Jess, who will be 3 times as old as Pam is now. Meanwhile, Matt will be 9 years older than Jess in 8 years and twice as old as Rena in 4 years. How old are Pam, Rena, Jess, and Matt currently?<|im_end|>
<|im_start|>assistant
<Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If Pam is currently twice as young as Rena is, and in 12 years, Rena will be 7 years older than Jess, who will be 3 times as old as Pam is now. Meanwhile, Matt will be 9 years older than Jess in 8 years and twice as old as Rena in 4 years. How old are Pam, Rena, Jess, and Matt currently?<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

Question: If Pam is currently twice as young as Rena is, and in 12 years, Rena will be 7 years older than Jess, who will be 3 times as old as Pam is now. Meanwhile, Matt will be 9 years older than Jess in 8 years and twice as old as Rena in 4 years. How old are Pam, Rena, Jess, and Matt currently?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Emily is a certain age and her older sister, Rachel, is 24 years old. Rachel was 8 years old when Emily was half her age. How old is Emily now?<|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
Emily is a certain age and her older sister, Rachel, is 24 years old. Rachel was 8 years old when Emily was half her age. How old is Emily now?<|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
Emily is a certain age and her older sister, Rachel, is 24 years old. Rachel was 8 years old when Emily was half her age. How old is Emily now?<|im_end|>
<|im_start|>assistant
<Z_23><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Emily is a certain age and her older sister, Rachel, is 24 years old. Rachel was 8 years old when Emily was half her age. How old is Emily now?<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

Question: Emily is a certain age and her older sister, Rachel, is 24 years old. Rachel was 8 years old when Emily was half her age. How old is Emily now?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a=2005x+2006$, $b=2005x+2007$, $c=2005x+2008$, find $a^{2}+b^{2}+c^{2}-ab-ac-bc$.<|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 $a=2005x+2006$, $b=2005x+2007$, $c=2005x+2008$, find $a^{2}+b^{2}+c^{2}-ab-ac-bc$.<|im_end|>
<|im_start|>assistant
<Z_49><ANSWER>

Question: Given $a=2005x+2006$, $b=2005x+2007$, $c=2005x+2008$, find $a^{2}+b^{2}+c^{2}-ab-ac-bc$.