<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.<|im_end|>
<|im_start|>assistant
<Z_477><Z_264><Z_391><Z_194><Z_250><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.<|im_end|>
<|im_start|>assistant
<Z_193><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.<|im_end|>
<|im_start|>assistant
<Z_190><Z_257><Z_124><Z_344><Z_225><ANSWER>

Question: Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( S = \{ x \in \mathbf{R} \mid x^{2} + 2x + a = 0 \} \), if \( -1 \in S \), find the real number \( a \).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( S = \{ x \in \mathbf{R} \mid x^{2} + 2x + a = 0 \} \), if \( -1 \in S \), find the real number \( a \).<|im_end|>
<|im_start|>assistant
<Z_369><Z_486><Z_503><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( S = \{ x \in \mathbf{R} \mid x^{2} + 2x + a = 0 \} \), if \( -1 \in S \), find the real number \( a \).<|im_end|>
<|im_start|>assistant
<Z_129><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set \( S = \{ x \in \mathbf{R} \mid x^{2} + 2x + a = 0 \} \), if \( -1 \in S \), find the real number \( a \).<|im_end|>
<|im_start|>assistant
<Z_214><Z_173><Z_115><ANSWER>

Question: Given the set \( S = \{ x \in \mathbf{R} \mid x^{2} + 2x + a = 0 \} \), if \( -1 \in S \), find the real number \( a \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
<|im_end|>
<|im_start|>assistant
<Z_369><Z_509><Z_470><Z_453><Z_435><Z_341><Z_71><Z_389><Z_419><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
<|im_end|>
<|im_start|>assistant
<Z_364><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
<|im_end|>
<|im_start|>assistant
<Z_260><Z_446><Z_92><Z_189><Z_435><Z_266><Z_5><Z_492><Z_28><ANSWER>

Question: A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John started collecting coins. He collected 12 coins on the first day. Each subsequent day, he collected twice as many coins as he had collected the previous day. If he collected coins for 4 consecutive days, what was the average number of coins collected per day?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John started collecting coins. He collected 12 coins on the first day. Each subsequent day, he collected twice as many coins as he had collected the previous day. If he collected coins for 4 consecutive days, what was the average number of coins collected per day?<|im_end|>
<|im_start|>assistant
<Z_24><Z_240><Z_92><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John started collecting coins. He collected 12 coins on the first day. Each subsequent day, he collected twice as many coins as he had collected the previous day. If he collected coins for 4 consecutive days, what was the average number of coins collected per day?<|im_end|>
<|im_start|>assistant
<Z_427><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
John started collecting coins. He collected 12 coins on the first day. Each subsequent day, he collected twice as many coins as he had collected the previous day. If he collected coins for 4 consecutive days, what was the average number of coins collected per day?<|im_end|>
<|im_start|>assistant
<Z_311><Z_442><Z_230><Z_474><ANSWER>

Question: John started collecting coins. He collected 12 coins on the first day. Each subsequent day, he collected twice as many coins as he had collected the previous day. If he collected coins for 4 consecutive days, what was the average number of coins collected per day?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Part of the graph of the polynomial function $g(x) = ax^3 + bx^2 + cx + d$ is given. The roots of $g(x)$ include $-2$ and $1$, and the function passes through the point $(0, 3)$. Additionally, the derivative $g'(1) = 0$. Determine the value of $b$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Part of the graph of the polynomial function $g(x) = ax^3 + bx^2 + cx + d$ is given. The roots of $g(x)$ include $-2$ and $1$, and the function passes through the point $(0, 3)$. Additionally, the derivative $g'(1) = 0$. Determine the value of $b$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_357><Z_335><Z_140><Z_61><Z_497><Z_201><Z_116><Z_70><Z_157><Z_498><Z_446><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Part of the graph of the polynomial function $g(x) = ax^3 + bx^2 + cx + d$ is given. The roots of $g(x)$ include $-2$ and $1$, and the function passes through the point $(0, 3)$. Additionally, the derivative $g'(1) = 0$. Determine the value of $b$.<|im_end|>
<|im_start|>assistant
<Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Part of the graph of the polynomial function $g(x) = ax^3 + bx^2 + cx + d$ is given. The roots of $g(x)$ include $-2$ and $1$, and the function passes through the point $(0, 3)$. Additionally, the derivative $g'(1) = 0$. Determine the value of $b$.<|im_end|>
<|im_start|>assistant
<Z_150><Z_163><Z_413><Z_73><Z_129><Z_498><Z_286><Z_443><Z_286><Z_49><Z_109><Z_402><ANSWER>

Question: Part of the graph of the polynomial function $g(x) = ax^3 + bx^2 + cx + d$ is given. The roots of $g(x)$ include $-2$ and $1$, and the function passes through the point $(0, 3)$. Additionally, the derivative $g'(1) = 0$. Determine the value of $b$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jonathan, his younger sister, older brother, and youngest sister have full names containing 8+10, 5+10, 6+10, and 4+15 letters respectively. What is the total number of letters in all of their names together?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jonathan, his younger sister, older brother, and youngest sister have full names containing 8+10, 5+10, 6+10, and 4+15 letters respectively. What is the total number of letters in all of their names together?<|im_end|>
<|im_start|>assistant
<Z_312><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jonathan, his younger sister, older brother, and youngest sister have full names containing 8+10, 5+10, 6+10, and 4+15 letters respectively. What is the total number of letters in all of their names together?<|im_end|>
<|im_start|>assistant
<Z_453><ANSWER>

Question: Jonathan, his younger sister, older brother, and youngest sister have full names containing 8+10, 5+10, 6+10, and 4+15 letters respectively. What is the total number of letters in all of their names together?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 15 students of a class is 15 years. Out of these, the average age of 3 students is 14 years and that of the other 11 students is 16 years. What is the age of the 15th student?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 15 students of a class is 15 years. Out of these, the average age of 3 students is 14 years and that of the other 11 students is 16 years. What is the age of the 15th student?<|im_end|>
<|im_start|>assistant
<Z_369><Z_465><Z_456><Z_123><Z_393><Z_106><Z_392><Z_508><Z_253><Z_64><Z_157><Z_421><Z_491><Z_267><Z_43><Z_71><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 15 students of a class is 15 years. Out of these, the average age of 3 students is 14 years and that of the other 11 students is 16 years. What is the age of the 15th student?<|im_end|>
<|im_start|>assistant
<Z_492><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of 15 students of a class is 15 years. Out of these, the average age of 3 students is 14 years and that of the other 11 students is 16 years. What is the age of the 15th student?<|im_end|>
<|im_start|>assistant
<Z_498><Z_348><Z_446><Z_511><Z_348><Z_400><Z_2><Z_209><Z_284><Z_323><Z_382><Z_315><Z_492><Z_335><Z_195><Z_86><Z_380><ANSWER>

Question: The average age of 15 students of a class is 15 years. Out of these, the average age of 3 students is 14 years and that of the other 11 students is 16 years. What is the age of the 15th student?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sequence, the first term is \(a_1 = 1007\) and the second term is \(a_2 = 1008.\) The values of the remaining terms satisfy the condition:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\] for all \(n \geq 1.\) Determine \(a_{1000}.\)<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sequence, the first term is \(a_1 = 1007\) and the second term is \(a_2 = 1008.\) The values of the remaining terms satisfy the condition:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\] for all \(n \geq 1.\) Determine \(a_{1000}.\)<|im_end|>
<|im_start|>assistant
<Z_369><Z_194><Z_501><Z_459><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sequence, the first term is \(a_1 = 1007\) and the second term is \(a_2 = 1008.\) The values of the remaining terms satisfy the condition:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\] for all \(n \geq 1.\) Determine \(a_{1000}.\)<|im_end|>
<|im_start|>assistant
<Z_371><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a sequence, the first term is \(a_1 = 1007\) and the second term is \(a_2 = 1008.\) The values of the remaining terms satisfy the condition:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\] for all \(n \geq 1.\) Determine \(a_{1000}.\)<|im_end|>
<|im_start|>assistant
<Z_240><Z_427><Z_206><Z_102><ANSWER>

Question:
In a sequence, the first term is \(a_1 = 1007\) and the second term is \(a_2 = 1008.\) The values of the remaining terms satisfy the condition:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\] for all \(n \geq 1.\) Determine \(a_{1000}.\)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A square has sides that measure 15 cm. A rectangle has a length of 18 cm. The perimeter of the square is equal to the perimeter of the rectangle. What is the area of the rectangle?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A square has sides that measure 15 cm. A rectangle has a length of 18 cm. The perimeter of the square is equal to the perimeter of the rectangle. What is the area of the rectangle?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A square has sides that measure 15 cm. A rectangle has a length of 18 cm. The perimeter of the square is equal to the perimeter of the rectangle. What is the area of the rectangle?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A square has sides that measure 15 cm. A rectangle has a length of 18 cm. The perimeter of the square is equal to the perimeter of the rectangle. What is the area of the rectangle?<|im_end|>
<|im_start|>assistant
<Z_141><ANSWER>

Question: A square has sides that measure 15 cm. A rectangle has a length of 18 cm. The perimeter of the square is equal to the perimeter of the rectangle. What is the area of the rectangle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ways to arrange the letters of the word BANANA.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ways to arrange the letters of the word BANANA.<|im_end|>
<|im_start|>assistant
<Z_477><Z_503><Z_239><Z_132><Z_362><Z_504><Z_202><Z_102><Z_274><Z_118><Z_205><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ways to arrange the letters of the word BANANA.<|im_end|>
<|im_start|>assistant
<Z_429><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ways to arrange the letters of the word BANANA.<|im_end|>
<|im_start|>assistant
<Z_71><Z_4><Z_327><Z_176><Z_378><Z_184><Z_403><Z_88><Z_52><Z_385><Z_452><ANSWER>

Question: Determine the number of ways to arrange the letters of the word BANANA.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{15}{12} + 10$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{15}{12} + 10$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_140><Z_149><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{15}{12} + 10$.<|im_end|>
<|im_start|>assistant
<Z_221><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{15}{12} + 10$.<|im_end|>
<|im_start|>assistant
<Z_474><Z_402><Z_17><ANSWER>

Question: Compute $\binom{15}{12} + 10$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{9}{5}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{9}{5}$.<|im_end|>
<|im_start|>assistant
<Z_314><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\binom{9}{5}$.<|im_end|>
<|im_start|>assistant
<Z_130><ANSWER>

Question: Compute $\binom{9}{5}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Martin starts with 35 goldfish. Each week, a varying number of goldfish die: in the first week 10 goldfish die, in the second week 12 die, and each subsequent week 2 additional goldfish die over the previous week. Martin purchases new goldfish every week: in the first week he purchases 5, in the second week he purchases 7, and each subsequent week he purchases 2 additional goldfish over the previous week. How many goldfish will Martin have after 15 weeks?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Martin starts with 35 goldfish. Each week, a varying number of goldfish die: in the first week 10 goldfish die, in the second week 12 die, and each subsequent week 2 additional goldfish die over the previous week. Martin purchases new goldfish every week: in the first week he purchases 5, in the second week he purchases 7, and each subsequent week he purchases 2 additional goldfish over the previous week. How many goldfish will Martin have after 15 weeks?<|im_end|>
<|im_start|>assistant
<Z_369><Z_75><Z_229><Z_434><Z_99><Z_71><Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Martin starts with 35 goldfish. Each week, a varying number of goldfish die: in the first week 10 goldfish die, in the second week 12 die, and each subsequent week 2 additional goldfish die over the previous week. Martin purchases new goldfish every week: in the first week he purchases 5, in the second week he purchases 7, and each subsequent week he purchases 2 additional goldfish over the previous week. How many goldfish will Martin have after 15 weeks?<|im_end|>
<|im_start|>assistant
<Z_230><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Martin starts with 35 goldfish. Each week, a varying number of goldfish die: in the first week 10 goldfish die, in the second week 12 die, and each subsequent week 2 additional goldfish die over the previous week. Martin purchases new goldfish every week: in the first week he purchases 5, in the second week he purchases 7, and each subsequent week he purchases 2 additional goldfish over the previous week. How many goldfish will Martin have after 15 weeks?<|im_end|>
<|im_start|>assistant
<Z_200><Z_506><Z_64><Z_428><Z_130><Z_367><Z_449><ANSWER>

Question: Martin starts with 35 goldfish. Each week, a varying number of goldfish die: in the first week 10 goldfish die, in the second week 12 die, and each subsequent week 2 additional goldfish die over the previous week. Martin purchases new goldfish every week: in the first week he purchases 5, in the second week he purchases 7, and each subsequent week he purchases 2 additional goldfish over the previous week. How many goldfish will Martin have after 15 weeks?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?<|im_end|>
<|im_start|>assistant
<Z_24><Z_394><Z_100><Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?<|im_end|>
<|im_start|>assistant
<Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?<|im_end|>
<|im_start|>assistant
<Z_128><Z_304><Z_255><Z_204><ANSWER>

Question: Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Heeju's school has 500 students. Among them, 289 students like ramen and 337 students like bread. If there are 56 students who do not like both ramen and bread, find how many students like both ramen and bread.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Heeju's school has 500 students. Among them, 289 students like ramen and 337 students like bread. If there are 56 students who do not like both ramen and bread, find how many students like both ramen and bread.<|im_end|>
<|im_start|>assistant
<Z_24><Z_210><Z_118><Z_318><Z_134><Z_264><Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Heeju's school has 500 students. Among them, 289 students like ramen and 337 students like bread. If there are 56 students who do not like both ramen and bread, find how many students like both ramen and bread.<|im_end|>
<|im_start|>assistant
<Z_412><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Heeju's school has 500 students. Among them, 289 students like ramen and 337 students like bread. If there are 56 students who do not like both ramen and bread, find how many students like both ramen and bread.<|im_end|>
<|im_start|>assistant
<Z_417><Z_86><Z_406><Z_508><Z_287><Z_46><Z_270><ANSWER>

Question: Heeju's school has 500 students. Among them, 289 students like ramen and 337 students like bread. If there are 56 students who do not like both ramen and bread, find how many students like both ramen and bread.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base k of the number system, if ( 524 ) 8 = ( 664 ) k?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base k of the number system, if ( 524 ) 8 = ( 664 ) k?<|im_end|>
<|im_start|>assistant
<Z_477><Z_192><Z_245><Z_348><Z_263><Z_139><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base k of the number system, if ( 524 ) 8 = ( 664 ) k?<|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
Find the base k of the number system, if ( 524 ) 8 = ( 664 ) k?<|im_end|>
<|im_start|>assistant
<Z_60><Z_65><Z_146><Z_65><Z_325><Z_57><ANSWER>

Question: Find the base k of the number system, if ( 524 ) 8 = ( 664 ) k?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if 6 < x < 10 < y < 17 , then what is the greatest possible positive integer difference of x and y ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if 6 < x < 10 < y < 17 , then what is the greatest possible positive integer difference of x and y ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_274><Z_157><Z_310><Z_348><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if 6 < x < 10 < y < 17 , then what is the greatest possible positive integer difference of x and y ?<|im_end|>
<|im_start|>assistant
<Z_74><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if 6 < x < 10 < y < 17 , then what is the greatest possible positive integer difference of x and y ?<|im_end|>
<|im_start|>assistant
<Z_511><Z_421><Z_383><Z_179><Z_187><ANSWER>

Question: if 6 < x < 10 < y < 17 , then what is the greatest possible positive integer difference of x and y ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?<|im_end|>
<|im_start|>assistant
<Z_477><Z_229><Z_357><Z_260><Z_193><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?<|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
Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?<|im_end|>
<|im_start|>assistant
<Z_218><Z_450><Z_150><Z_89><Z_125><Z_289><ANSWER>

Question: Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the difference between the quotients of two divisions 32.5 / 1.3 and 60.8 / 7.6.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the difference between the quotients of two divisions 32.5 / 1.3 and 60.8 / 7.6.<|im_end|>
<|im_start|>assistant
<Z_477><Z_426><Z_55><Z_366><Z_361><Z_91><Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the difference between the quotients of two divisions 32.5 / 1.3 and 60.8 / 7.6.<|im_end|>
<|im_start|>assistant
<Z_296><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the difference between the quotients of two divisions 32.5 / 1.3 and 60.8 / 7.6.<|im_end|>
<|im_start|>assistant
<Z_302><Z_148><Z_22><Z_334><Z_44><Z_473><Z_214><ANSWER>

Question: Find the difference between the quotients of two divisions 32.5 / 1.3 and 60.8 / 7.6.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(3^x) = 4x\log_{2}3 + 233$, find the value of $f(2) + f(4) + f(8) + \ldots + f(2^8)$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(3^x) = 4x\log_{2}3 + 233$, find the value of $f(2) + f(4) + f(8) + \ldots + f(2^8)$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_294><Z_55><Z_68><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(3^x) = 4x\log_{2}3 + 233$, find the value of $f(2) + f(4) + f(8) + \ldots + f(2^8)$.<|im_end|>
<|im_start|>assistant
<Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(3^x) = 4x\log_{2}3 + 233$, find the value of $f(2) + f(4) + f(8) + \ldots + f(2^8)$.<|im_end|>
<|im_start|>assistant
<Z_88><Z_26><Z_500><Z_6><Z_324><ANSWER>

Question: Given $f(3^x) = 4x\log_{2}3 + 233$, find the value of $f(2) + f(4) + f(8) + \ldots + f(2^8)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing a certain amount. The weight of the new person might be 80 kg. What was the weight of the person who was replaced?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing a certain amount. The weight of the new person might be 80 kg. What was the weight of the person who was replaced?<|im_end|>
<|im_start|>assistant
<Z_24><Z_47><Z_173><Z_246><Z_144><Z_159><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing a certain amount. The weight of the new person might be 80 kg. What was the weight of the person who was replaced?<|im_end|>
<|im_start|>assistant
<Z_145><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing a certain amount. The weight of the new person might be 80 kg. What was the weight of the person who was replaced?<|im_end|>
<|im_start|>assistant
<Z_127><Z_38><Z_253><Z_224><Z_12><Z_46><Z_384><ANSWER>

Question: The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing a certain amount. The weight of the new person might be 80 kg. What was the weight of the person who was replaced?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe has a daughter called Jane. The sum of their ages is equal to 54 and the difference between their ages is 22. How old is Joe?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe has a daughter called Jane. The sum of their ages is equal to 54 and the difference between their ages is 22. How old is Joe?<|im_end|>
<|im_start|>assistant
<Z_369><Z_127><Z_101><Z_335><Z_274><Z_246><Z_264><Z_346><Z_322><Z_73><Z_483><Z_168><Z_483><Z_300><Z_155><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe has a daughter called Jane. The sum of their ages is equal to 54 and the difference between their ages is 22. How old is Joe?<|im_end|>
<|im_start|>assistant
<Z_500><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe has a daughter called Jane. The sum of their ages is equal to 54 and the difference between their ages is 22. How old is Joe?<|im_end|>
<|im_start|>assistant
<Z_281><Z_62><Z_249><Z_277><Z_507><Z_332><Z_0><Z_126><Z_341><Z_175><Z_142><Z_216><Z_13><Z_414><Z_147><Z_105><ANSWER>

Question: Joe has a daughter called Jane. The sum of their ages is equal to 54 and the difference between their ages is 22. How old is Joe?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive three-digit integers are divisible by both 13 and 5?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive three-digit integers are divisible by both 13 and 5?<|im_end|>
<|im_start|>assistant
<Z_24><Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive three-digit integers are divisible by both 13 and 5?<|im_end|>
<|im_start|>assistant
<Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive three-digit integers are divisible by both 13 and 5?<|im_end|>
<|im_start|>assistant
<Z_61><Z_181><ANSWER>

Question: How many positive three-digit integers are divisible by both 13 and 5?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3x^{2m}y^{m}$ and $x^{4-n}y^{n-1}$ are like terms, then $m+n=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3x^{2m}y^{m}$ and $x^{4-n}y^{n-1}$ are like terms, then $m+n=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_369><Z_437><Z_390><Z_315><Z_227><Z_400><Z_264><Z_265><Z_178><Z_260><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3x^{2m}y^{m}$ and $x^{4-n}y^{n-1}$ are like terms, then $m+n=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3x^{2m}y^{m}$ and $x^{4-n}y^{n-1}$ are like terms, then $m+n=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_136><Z_47><Z_303><Z_160><Z_369><Z_68><Z_218><Z_266><Z_281><Z_11><ANSWER>

Question: If $3x^{2m}y^{m}$ and $x^{4-n}y^{n-1}$ are like terms, then $m+n=$ \_\_\_\_\_\_.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x < -2, \\
10 - 3x &\text{if }x \geq -2.
\end{cases}
\]
Find \( f(-1) \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x < -2, \\
10 - 3x &\text{if }x \geq -2.
\end{cases}
\]
Find \( f(-1) \).<|im_end|>
<|im_start|>assistant
<Z_477><Z_160><Z_102><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x < -2, \\
10 - 3x &\text{if }x \geq -2.
\end{cases}
\]
Find \( f(-1) \).<|im_end|>
<|im_start|>assistant
<Z_481><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x < -2, \\
10 - 3x &\text{if }x \geq -2.
\end{cases}
\]
Find \( f(-1) \).<|im_end|>
<|im_start|>assistant
<Z_262><Z_482><Z_394><ANSWER>

Question:
Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x < -2, \\
10 - 3x &\text{if }x \geq -2.
\end{cases}
\]
Find \( f(-1) \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
She has some crayons which she will place in crayon boxes. Every box can contain 8 crayons. She needs 10 boxes. How many crayons does she have?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
She has some crayons which she will place in crayon boxes. Every box can contain 8 crayons. She needs 10 boxes. How many crayons does she have?<|im_end|>
<|im_start|>assistant
<Z_24><Z_373><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
She has some crayons which she will place in crayon boxes. Every box can contain 8 crayons. She needs 10 boxes. How many crayons does she have?<|im_end|>
<|im_start|>assistant
<Z_461><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
She has some crayons which she will place in crayon boxes. Every box can contain 8 crayons. She needs 10 boxes. How many crayons does she have?<|im_end|>
<|im_start|>assistant
<Z_373><Z_20><ANSWER>

Question: She has some crayons which she will place in crayon boxes. Every box can contain 8 crayons. She needs 10 boxes. How many crayons does she have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the sequence where the numbers are of the form $a_n=150 + 3n^2$, where $n = 1, 2, 3, \ldots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the sequence where the numbers are of the form $a_n=150 + 3n^2$, where $n = 1, 2, 3, \ldots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.<|im_end|>
<|im_start|>assistant
<Z_369><Z_408><Z_482><Z_425><Z_255><Z_494><Z_144><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the sequence where the numbers are of the form $a_n=150 + 3n^2$, where $n = 1, 2, 3, \ldots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.<|im_end|>
<|im_start|>assistant
<Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the sequence where the numbers are of the form $a_n=150 + 3n^2$, where $n = 1, 2, 3, \ldots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.<|im_end|>
<|im_start|>assistant
<Z_458><Z_430><Z_390><Z_117><Z_116><Z_116><Z_461><ANSWER>

Question: Consider the sequence where the numbers are of the form $a_n=150 + 3n^2$, where $n = 1, 2, 3, \ldots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?<|im_end|>
<|im_start|>assistant
<Z_24><Z_273><Z_237><Z_502><Z_289><Z_376><Z_318><Z_240><Z_463><Z_319><Z_335><Z_213><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?<|im_end|>
<|im_start|>assistant
<Z_349><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?<|im_end|>
<|im_start|>assistant
<Z_21><Z_501><Z_303><Z_388><Z_494><Z_258><Z_457><Z_189><Z_375><Z_143><Z_106><Z_495><ANSWER>

Question: There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $ABCD$ be an isosceles trapezoid with bases $AB = 80$ and $CD = 17$. Suppose $AD = BC = x$, and a circle with center on line segment $\overline{AB}$ is tangent to $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, find $m^2$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $ABCD$ be an isosceles trapezoid with bases $AB = 80$ and $CD = 17$. Suppose $AD = BC = x$, and a circle with center on line segment $\overline{AB}$ is tangent to $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, find $m^2$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_232><Z_461><Z_201><Z_391><Z_509><Z_286><Z_288><Z_63><Z_98><Z_421><Z_364><Z_322><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $ABCD$ be an isosceles trapezoid with bases $AB = 80$ and $CD = 17$. Suppose $AD = BC = x$, and a circle with center on line segment $\overline{AB}$ is tangent to $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, find $m^2$.<|im_end|>
<|im_start|>assistant
<Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $ABCD$ be an isosceles trapezoid with bases $AB = 80$ and $CD = 17$. Suppose $AD = BC = x$, and a circle with center on line segment $\overline{AB}$ is tangent to $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, find $m^2$.<|im_end|>
<|im_start|>assistant
<Z_503><Z_452><Z_177><Z_189><Z_103><Z_276><Z_161><Z_117><Z_166><Z_7><Z_83><Z_153><Z_93><ANSWER>

Question: Let $ABCD$ be an isosceles trapezoid with bases $AB = 80$ and $CD = 17$. Suppose $AD = BC = x$, and a circle with center on line segment $\overline{AB}$ is tangent to $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, find $m^2$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a pupil ' s marks were wrongly entered as 85 instead of 45 . due to that the average marks for the class got increased by half . the number of pupils in the class is :<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a pupil ' s marks were wrongly entered as 85 instead of 45 . due to that the average marks for the class got increased by half . the number of pupils in the class is :<|im_end|>
<|im_start|>assistant
<Z_369><Z_437><Z_173><Z_330><Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a pupil ' s marks were wrongly entered as 85 instead of 45 . due to that the average marks for the class got increased by half . the number of pupils in the class is :<|im_end|>
<|im_start|>assistant
<Z_189><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a pupil ' s marks were wrongly entered as 85 instead of 45 . due to that the average marks for the class got increased by half . the number of pupils in the class is :<|im_end|>
<|im_start|>assistant
<Z_87><Z_226><Z_224><Z_446><Z_290><ANSWER>

Question: a pupil ' s marks were wrongly entered as 85 instead of 45 . due to that the average marks for the class got increased by half . the number of pupils in the class is :


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Sean Li*<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Sean Li*<|im_end|>
<|im_start|>assistant
<Z_24><Z_129><Z_72><Z_394><Z_286><Z_488><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Sean Li*<|im_end|>
<|im_start|>assistant
<Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Sean Li*<|im_end|>
<|im_start|>assistant
<Z_400><Z_475><Z_193><Z_184><Z_313><Z_332><ANSWER>

Question:
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Sean Li*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after some time. The second blast occurred exactly 30 mins after the first. He was 4950 meters from the site when he heard the second blast. (speed of sound = 330 m/s) How long did it take for him to hear the second blast?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after some time. The second blast occurred exactly 30 mins after the first. He was 4950 meters from the site when he heard the second blast. (speed of sound = 330 m/s) How long did it take for him to hear the second blast?<|im_end|>
<|im_start|>assistant
<Z_369><Z_420><Z_462><Z_194><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after some time. The second blast occurred exactly 30 mins after the first. He was 4950 meters from the site when he heard the second blast. (speed of sound = 330 m/s) How long did it take for him to hear the second blast?<|im_end|>
<|im_start|>assistant
<Z_393><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after some time. The second blast occurred exactly 30 mins after the first. He was 4950 meters from the site when he heard the second blast. (speed of sound = 330 m/s) How long did it take for him to hear the second blast?<|im_end|>
<|im_start|>assistant
<Z_446><Z_230><Z_80><Z_398><Z_113><ANSWER>

Question: A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after some time. The second blast occurred exactly 30 mins after the first. He was 4950 meters from the site when he heard the second blast. (speed of sound = 330 m/s) How long did it take for him to hear the second blast?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maya loves to read books. Last week she read 5 books. Each book had a certain number of pages of text. This week she read twice as much. Maya read a total of 4500 pages. How many pages did each book have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maya loves to read books. Last week she read 5 books. Each book had a certain number of pages of text. This week she read twice as much. Maya read a total of 4500 pages. How many pages did each book have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_389><Z_484><Z_397><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maya loves to read books. Last week she read 5 books. Each book had a certain number of pages of text. This week she read twice as much. Maya read a total of 4500 pages. How many pages did each book have?<|im_end|>
<|im_start|>assistant
<Z_218><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maya loves to read books. Last week she read 5 books. Each book had a certain number of pages of text. This week she read twice as much. Maya read a total of 4500 pages. How many pages did each book have?<|im_end|>
<|im_start|>assistant
<Z_260><Z_391><Z_348><Z_140><ANSWER>

Question: Maya loves to read books. Last week she read 5 books. Each book had a certain number of pages of text. This week she read twice as much. Maya read a total of 4500 pages. How many pages did each book have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eggs are sold in packages of some size or 11 only. Doris bought 79 eggs exactly, and she bought 5 large packs. What is the size of the smaller pack?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eggs are sold in packages of some size or 11 only. Doris bought 79 eggs exactly, and she bought 5 large packs. What is the size of the smaller pack?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eggs are sold in packages of some size or 11 only. Doris bought 79 eggs exactly, and she bought 5 large packs. What is the size of the smaller pack?<|im_end|>
<|im_start|>assistant
<Z_367><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eggs are sold in packages of some size or 11 only. Doris bought 79 eggs exactly, and she bought 5 large packs. What is the size of the smaller pack?<|im_end|>
<|im_start|>assistant
<Z_250><ANSWER>

Question: Eggs are sold in packages of some size or 11 only. Doris bought 79 eggs exactly, and she bought 5 large packs. What is the size of the smaller pack?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jim buys a package of some car washes. Since he bought a package, he only has to pay 60%. A carwash normally costs 15 dollars. He paid $180 for the package. How many car washes were in the package?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jim buys a package of some car washes. Since he bought a package, he only has to pay 60%. A carwash normally costs 15 dollars. He paid $180 for the package. How many car washes were in the package?<|im_end|>
<|im_start|>assistant
<Z_369><Z_487><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jim buys a package of some car washes. Since he bought a package, he only has to pay 60%. A carwash normally costs 15 dollars. He paid $180 for the package. How many car washes were in the package?<|im_end|>
<|im_start|>assistant
<Z_281><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jim buys a package of some car washes. Since he bought a package, he only has to pay 60%. A carwash normally costs 15 dollars. He paid $180 for the package. How many car washes were in the package?<|im_end|>
<|im_start|>assistant
<Z_488><Z_34><Z_358><ANSWER>

Question: Jim buys a package of some car washes. Since he bought a package, he only has to pay 60%. A carwash normally costs 15 dollars. He paid $180 for the package. How many car washes were in the package?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thin rectangular slab of potato was cut into two pieces for an osmosis lab. One piece is 50 mm greater than the other. The length of the other piece of the potato after it is cut is 275 mm. What was the length of the original uncut slab?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thin rectangular slab of potato was cut into two pieces for an osmosis lab. One piece is 50 mm greater than the other. The length of the other piece of the potato after it is cut is 275 mm. What was the length of the original uncut slab?<|im_end|>
<|im_start|>assistant
<Z_24><Z_446><Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thin rectangular slab of potato was cut into two pieces for an osmosis lab. One piece is 50 mm greater than the other. The length of the other piece of the potato after it is cut is 275 mm. What was the length of the original uncut slab?<|im_end|>
<|im_start|>assistant
<Z_315><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thin rectangular slab of potato was cut into two pieces for an osmosis lab. One piece is 50 mm greater than the other. The length of the other piece of the potato after it is cut is 275 mm. What was the length of the original uncut slab?<|im_end|>
<|im_start|>assistant
<Z_1><Z_443><Z_406><ANSWER>

Question: A thin rectangular slab of potato was cut into two pieces for an osmosis lab. One piece is 50 mm greater than the other. The length of the other piece of the potato after it is cut is 275 mm. What was the length of the original uncut slab?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given four members of a group -- Alice, Bob, Carol, and Dave -- in how many ways can any three of these four be chosen and assigned to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given four members of a group -- Alice, Bob, Carol, and Dave -- in how many ways can any three of these four be chosen and assigned to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job?<|im_end|>
<|im_start|>assistant
<Z_477><Z_250><Z_151><Z_273><Z_289><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given four members of a group -- Alice, Bob, Carol, and Dave -- in how many ways can any three of these four be chosen and assigned to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job?<|im_end|>
<|im_start|>assistant
<Z_138><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given four members of a group -- Alice, Bob, Carol, and Dave -- in how many ways can any three of these four be chosen and assigned to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job?<|im_end|>
<|im_start|>assistant
<Z_210><Z_159><Z_192><Z_208><Z_6><Z_447><ANSWER>

Question: Given four members of a group -- Alice, Bob, Carol, and Dave -- in how many ways can any three of these four be chosen and assigned to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the configuration below, the largest circle has a radius of 10 meters. Three identical smaller circles are placed inside and are touching each other as shown, forming a straight line from one point on the circumference of the larger circle to another point directly opposite. What is the radius in meters of one of the three smaller circles?

[asy]
size(3cm,3cm);
draw(Circle((0,0),1));
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((0,0),3));
[/asy]<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the configuration below, the largest circle has a radius of 10 meters. Three identical smaller circles are placed inside and are touching each other as shown, forming a straight line from one point on the circumference of the larger circle to another point directly opposite. What is the radius in meters of one of the three smaller circles?

[asy]
size(3cm,3cm);
draw(Circle((0,0),1));
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((0,0),3));
[/asy]<|im_end|>
<|im_start|>assistant
<Z_477><Z_340><Z_77><Z_335><Z_71><Z_284><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the configuration below, the largest circle has a radius of 10 meters. Three identical smaller circles are placed inside and are touching each other as shown, forming a straight line from one point on the circumference of the larger circle to another point directly opposite. What is the radius in meters of one of the three smaller circles?

[asy]
size(3cm,3cm);
draw(Circle((0,0),1));
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((0,0),3));
[/asy]<|im_end|>
<|im_start|>assistant
<Z_127><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the configuration below, the largest circle has a radius of 10 meters. Three identical smaller circles are placed inside and are touching each other as shown, forming a straight line from one point on the circumference of the larger circle to another point directly opposite. What is the radius in meters of one of the three smaller circles?

[asy]
size(3cm,3cm);
draw(Circle((0,0),1));
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((0,0),3));
[/asy]<|im_end|>
<|im_start|>assistant
<Z_142><Z_333><Z_153><Z_233><Z_261><Z_476><ANSWER>

Question:
In the configuration below, the largest circle has a radius of 10 meters. Three identical smaller circles are placed inside and are touching each other as shown, forming a straight line from one point on the circumference of the larger circle to another point directly opposite. What is the radius in meters of one of the three smaller circles?

[asy]
size(3cm,3cm);
draw(Circle((0,0),1));
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((0,0),3));
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train starts from Amritsar to Bombay at 9 am. It reaches the destination after 3 days at 9:30 am. Every day a train starts. How many trains does it come across on the way?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train starts from Amritsar to Bombay at 9 am. It reaches the destination after 3 days at 9:30 am. Every day a train starts. How many trains does it come across on the way?<|im_end|>
<|im_start|>assistant
<Z_369><Z_236><Z_316><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train starts from Amritsar to Bombay at 9 am. It reaches the destination after 3 days at 9:30 am. Every day a train starts. How many trains does it come across on the way?<|im_end|>
<|im_start|>assistant
<Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train starts from Amritsar to Bombay at 9 am. It reaches the destination after 3 days at 9:30 am. Every day a train starts. How many trains does it come across on the way?<|im_end|>
<|im_start|>assistant
<Z_112><Z_37><Z_31><ANSWER>

Question: A train starts from Amritsar to Bombay at 9 am. It reaches the destination after 3 days at 9:30 am. Every day a train starts. How many trains does it come across on the way?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At Eastside Junior High, the $150$ students who participate in the math club meet in the evening to discuss math problems and each consumes on average $3$ cookies. Emma and Oliver are baking Emma's Excellent Cookie Squares this year. Their recipe, which makes a batch of $20$ cookies, includes these items:

$\bullet$ $2$ cups of flour

$\bullet$ $3$ eggs

$\bullet$ $4$ tablespoons butter

$\bullet$ $1$ cup sugar

$\bullet$ $1$ package of chocolate chips

Due to another event overlapping, attendance is expected to decrease by $30\%$. How many recipes of cookies should Emma and Oliver prepare for their event?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At Eastside Junior High, the $150$ students who participate in the math club meet in the evening to discuss math problems and each consumes on average $3$ cookies. Emma and Oliver are baking Emma's Excellent Cookie Squares this year. Their recipe, which makes a batch of $20$ cookies, includes these items:

$\bullet$ $2$ cups of flour

$\bullet$ $3$ eggs

$\bullet$ $4$ tablespoons butter

$\bullet$ $1$ cup sugar

$\bullet$ $1$ package of chocolate chips

Due to another event overlapping, attendance is expected to decrease by $30\%$. How many recipes of cookies should Emma and Oliver prepare for their event?<|im_end|>
<|im_start|>assistant
<Z_477><Z_425><Z_421><Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At Eastside Junior High, the $150$ students who participate in the math club meet in the evening to discuss math problems and each consumes on average $3$ cookies. Emma and Oliver are baking Emma's Excellent Cookie Squares this year. Their recipe, which makes a batch of $20$ cookies, includes these items:

$\bullet$ $2$ cups of flour

$\bullet$ $3$ eggs

$\bullet$ $4$ tablespoons butter

$\bullet$ $1$ cup sugar

$\bullet$ $1$ package of chocolate chips

Due to another event overlapping, attendance is expected to decrease by $30\%$. How many recipes of cookies should Emma and Oliver prepare for their event?<|im_end|>
<|im_start|>assistant
<Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At Eastside Junior High, the $150$ students who participate in the math club meet in the evening to discuss math problems and each consumes on average $3$ cookies. Emma and Oliver are baking Emma's Excellent Cookie Squares this year. Their recipe, which makes a batch of $20$ cookies, includes these items:

$\bullet$ $2$ cups of flour

$\bullet$ $3$ eggs

$\bullet$ $4$ tablespoons butter

$\bullet$ $1$ cup sugar

$\bullet$ $1$ package of chocolate chips

Due to another event overlapping, attendance is expected to decrease by $30\%$. How many recipes of cookies should Emma and Oliver prepare for their event?<|im_end|>
<|im_start|>assistant
<Z_316><Z_49><Z_215><Z_213><ANSWER>

Question:
At Eastside Junior High, the $150$ students who participate in the math club meet in the evening to discuss math problems and each consumes on average $3$ cookies. Emma and Oliver are baking Emma's Excellent Cookie Squares this year. Their recipe, which makes a batch of $20$ cookies, includes these items:

$\bullet$ $2$ cups of flour

$\bullet$ $3$ eggs

$\bullet$ $4$ tablespoons butter

$\bullet$ $1$ cup sugar

$\bullet$ $1$ package of chocolate chips

Due to another event overlapping, attendance is expected to decrease by $30\%$. How many recipes of cookies should Emma and Oliver prepare for their event?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Emily was playing a video game and had 42 lives. In a hard part of the game, she lost some lives. After getting 24 more lives in the next level, she had 41 lives. How many lives did Emily lose in the hard part of the game?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Emily was playing a video game and had 42 lives. In a hard part of the game, she lost some lives. After getting 24 more lives in the next level, she had 41 lives. How many lives did Emily lose in the hard part of the game?<|im_end|>
<|im_start|>assistant
<Z_477><Z_403><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Emily was playing a video game and had 42 lives. In a hard part of the game, she lost some lives. After getting 24 more lives in the next level, she had 41 lives. How many lives did Emily lose in the hard part of the game?<|im_end|>
<|im_start|>assistant
<Z_305><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Emily was playing a video game and had 42 lives. In a hard part of the game, she lost some lives. After getting 24 more lives in the next level, she had 41 lives. How many lives did Emily lose in the hard part of the game?<|im_end|>
<|im_start|>assistant
<Z_387><Z_63><ANSWER>

Question: Emily was playing a video game and had 42 lives. In a hard part of the game, she lost some lives. After getting 24 more lives in the next level, she had 41 lives. How many lives did Emily lose in the hard part of the game?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?<|im_end|>
<|im_start|>assistant
<Z_477><Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?<|im_end|>
<|im_start|>assistant
<Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?<|im_end|>
<|im_start|>assistant
<Z_489><Z_51><ANSWER>

Question: Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given space vectors $\overrightarrow{a}=(-2,1,3)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, then $x$ is equal to ______.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given space vectors $\overrightarrow{a}=(-2,1,3)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, then $x$ is equal to ______.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given space vectors $\overrightarrow{a}=(-2,1,3)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, then $x$ is equal to ______.<|im_end|>
<|im_start|>assistant
<Z_197><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given space vectors $\overrightarrow{a}=(-2,1,3)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, then $x$ is equal to ______.<|im_end|>
<|im_start|>assistant
<Z_466><ANSWER>

Question: Given space vectors $\overrightarrow{a}=(-2,1,3)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, then $x$ is equal to ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)$ function is defined as follows:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)$ function is defined as follows:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]<|im_end|>
<|im_start|>assistant
<Z_369><Z_100><Z_111><Z_139><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)$ function is defined as follows:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]<|im_end|>
<|im_start|>assistant
<Z_390><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)$ function is defined as follows:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]<|im_end|>
<|im_start|>assistant
<Z_350><Z_37><Z_293><Z_163><ANSWER>

Question:
Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)$ function is defined as follows:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Higher Secondary P7

If there exists a prime number  $p$  such that  $p+2q$  is prime for all positive integer  $q$  smaller than  $p$ , then  $p$  is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Higher Secondary P7

If there exists a prime number  $p$  such that  $p+2q$  is prime for all positive integer  $q$  smaller than  $p$ , then  $p$  is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.<|im_end|>
<|im_start|>assistant
<Z_477><Z_336><Z_322><Z_209><Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Higher Secondary P7

If there exists a prime number  $p$  such that  $p+2q$  is prime for all positive integer  $q$  smaller than  $p$ , then  $p$  is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.<|im_end|>
<|im_start|>assistant
<Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Higher Secondary P7

If there exists a prime number  $p$  such that  $p+2q$  is prime for all positive integer  $q$  smaller than  $p$ , then  $p$  is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.<|im_end|>
<|im_start|>assistant
<Z_158><Z_508><Z_123><Z_438><Z_386><ANSWER>

Question:
Higher Secondary P7

If there exists a prime number  $p$  such that  $p+2q$  is prime for all positive integer  $q$  smaller than  $p$ , then  $p$  is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and contains the point $(-2, 4)$. The equation of this line can be written in the form $y = mx + b$. What is the value of $m + b$?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and contains the point $(-2, 4)$. The equation of this line can be written in the form $y = mx + b$. What is the value of $m + b$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_322><Z_197><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and contains the point $(-2, 4)$. The equation of this line can be written in the form $y = mx + b$. What is the value of $m + b$?<|im_end|>
<|im_start|>assistant
<Z_381><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line has a slope of $5$ and contains the point $(-2, 4)$. The equation of this line can be written in the form $y = mx + b$. What is the value of $m + b$?<|im_end|>
<|im_start|>assistant
<Z_153><Z_473><Z_257><ANSWER>

Question: A line has a slope of $5$ and contains the point $(-2, 4)$. The equation of this line can be written in the form $y = mx + b$. What is the value of $m + b$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_416><Z_205><Z_252><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$.<|im_end|>
<|im_start|>assistant
<Z_335><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$.<|im_end|>
<|im_start|>assistant
<Z_364><Z_341><Z_232><Z_139><ANSWER>

Question: Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 4 men and 4 women be placed into two groups of two people and one group of four people, with the condition that there must be at least one man and one woman in the groups of two, and at least two men and two women in the group of four? Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 4 men and 4 women be placed into two groups of two people and one group of four people, with the condition that there must be at least one man and one woman in the groups of two, and at least two men and two women in the group of four? Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_477><Z_445><Z_252><Z_267><Z_291><Z_464><Z_501><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 4 men and 4 women be placed into two groups of two people and one group of four people, with the condition that there must be at least one man and one woman in the groups of two, and at least two men and two women in the group of four? Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_307><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 4 men and 4 women be placed into two groups of two people and one group of four people, with the condition that there must be at least one man and one woman in the groups of two, and at least two men and two women in the group of four? Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_375><Z_416><Z_304><Z_470><Z_82><Z_97><Z_475><ANSWER>

Question: In how many different ways can 4 men and 4 women be placed into two groups of two people and one group of four people, with the condition that there must be at least one man and one woman in the groups of two, and at least two men and two women in the group of four? Note that identically sized groups are indistinguishable.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers $M$ less than $500$ can be written as the sum of $k$ consecutive positive even integers from exactly 3 values of $k \geq 1$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers $M$ less than $500$ can be written as the sum of $k$ consecutive positive even integers from exactly 3 values of $k \geq 1$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_264><Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers $M$ less than $500$ can be written as the sum of $k$ consecutive positive even integers from exactly 3 values of $k \geq 1$?<|im_end|>
<|im_start|>assistant
<Z_257><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers $M$ less than $500$ can be written as the sum of $k$ consecutive positive even integers from exactly 3 values of $k \geq 1$?<|im_end|>
<|im_start|>assistant
<Z_205><Z_402><Z_410><ANSWER>

Question: How many integers $M$ less than $500$ can be written as the sum of $k$ consecutive positive even integers from exactly 3 values of $k \geq 1$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are four numbers 10, 11, 12, and 13. What is the difference between the largest number and the next largest number after that?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are four numbers 10, 11, 12, and 13. What is the difference between the largest number and the next largest number after that?<|im_end|>
<|im_start|>assistant
<Z_477><Z_209><Z_355><Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are four numbers 10, 11, 12, and 13. What is the difference between the largest number and the next largest number after that?<|im_end|>
<|im_start|>assistant
<Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are four numbers 10, 11, 12, and 13. What is the difference between the largest number and the next largest number after that?<|im_end|>
<|im_start|>assistant
<Z_85><Z_386><Z_132><Z_345><ANSWER>

Question: There are four numbers 10, 11, 12, and 13. What is the difference between the largest number and the next largest number after that?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of an adult class is 40 years. Some new students with an average age of 32 years join the class, therefore decreasing the average by 4 years. The original strength of the class was 8. How many new students joined the class?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of an adult class is 40 years. Some new students with an average age of 32 years join the class, therefore decreasing the average by 4 years. The original strength of the class was 8. How many new students joined the class?<|im_end|>
<|im_start|>assistant
<Z_24><Z_174><Z_427><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of an adult class is 40 years. Some new students with an average age of 32 years join the class, therefore decreasing the average by 4 years. The original strength of the class was 8. How many new students joined the class?<|im_end|>
<|im_start|>assistant
<Z_179><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of an adult class is 40 years. Some new students with an average age of 32 years join the class, therefore decreasing the average by 4 years. The original strength of the class was 8. How many new students joined the class?<|im_end|>
<|im_start|>assistant
<Z_158><Z_95><Z_294><ANSWER>

Question: The average age of an adult class is 40 years. Some new students with an average age of 32 years join the class, therefore decreasing the average by 4 years. The original strength of the class was 8. How many new students joined the class?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different $6\times 6$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row and each column is 0, and additionally the sum of the entries in every two adjacent rows and every two adjacent columns is also 0?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different $6\times 6$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row and each column is 0, and additionally the sum of the entries in every two adjacent rows and every two adjacent columns is also 0?<|im_end|>
<|im_start|>assistant
<Z_24><Z_179><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different $6\times 6$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row and each column is 0, and additionally the sum of the entries in every two adjacent rows and every two adjacent columns is also 0?<|im_end|>
<|im_start|>assistant
<Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different $6\times 6$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row and each column is 0, and additionally the sum of the entries in every two adjacent rows and every two adjacent columns is also 0?<|im_end|>
<|im_start|>assistant
<Z_281><Z_493><ANSWER>

Question: How many different $6\times 6$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row and each column is 0, and additionally the sum of the entries in every two adjacent rows and every two adjacent columns is also 0?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 10 days was finished 3 days earlier after some more men joined. The number of men employed was 15. How many additional men joined to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 10 days was finished 3 days earlier after some more men joined. The number of men employed was 15. How many additional men joined to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_477><Z_61><Z_451><Z_60><Z_335><Z_430><Z_104><Z_432><Z_98><Z_146><Z_38><Z_213><Z_72><Z_244><Z_138><Z_185><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 10 days was finished 3 days earlier after some more men joined. The number of men employed was 15. How many additional men joined to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 10 days was finished 3 days earlier after some more men joined. The number of men employed was 15. How many additional men joined to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_227><Z_362><Z_166><Z_11><Z_147><Z_21><Z_323><Z_385><Z_21><Z_375><Z_267><Z_22><Z_504><Z_244><Z_239><Z_476><ANSWER>

Question: A work which could be finished in 10 days was finished 3 days earlier after some more men joined. The number of men employed was 15. How many additional men joined to finish the work earlier?