<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\log_{16}(x-6) = \frac{1}{4}$, find $\frac{1}{\log_{x}2}$.<|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
If $\log_{16}(x-6) = \frac{1}{4}$, find $\frac{1}{\log_{x}2}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_187><Z_101><Z_164><Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\log_{16}(x-6) = \frac{1}{4}$, find $\frac{1}{\log_{x}2}$.<|im_end|>
<|im_start|>assistant
<Z_2><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\log_{16}(x-6) = \frac{1}{4}$, find $\frac{1}{\log_{x}2}$.<|im_end|>
<|im_start|>assistant
<Z_484><Z_64><Z_211><Z_382><Z_469><ANSWER>

Question: If $\log_{16}(x-6) = \frac{1}{4}$, find $\frac{1}{\log_{x}2}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the area of the figure bounded by the parabola \( y = -x^2 + 6x - 5 \) and the coordinate axes.<|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 the area of the figure bounded by the parabola \( y = -x^2 + 6x - 5 \) and the coordinate axes.<|im_end|>
<|im_start|>assistant
<Z_369><Z_351><Z_407><Z_437><Z_314><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the area of the figure bounded by the parabola \( y = -x^2 + 6x - 5 \) and the coordinate axes.<|im_end|>
<|im_start|>assistant
<Z_457><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the area of the figure bounded by the parabola \( y = -x^2 + 6x - 5 \) and the coordinate axes.<|im_end|>
<|im_start|>assistant
<Z_52><Z_107><Z_148><Z_28><Z_237><ANSWER>

Question: Compute the area of the figure bounded by the parabola \( y = -x^2 + 6x - 5 \) and the coordinate axes.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the sum $$1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 + 88888888 + 999999999$$ and determine what this total is congruent to modulo 9.<|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
Calculate the sum $$1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 + 88888888 + 999999999$$ and determine what this total is congruent to modulo 9.<|im_end|>
<|im_start|>assistant
<Z_369><Z_414><Z_496><Z_99><Z_394><Z_439><Z_273><Z_274><Z_378><Z_311><Z_415><Z_46><Z_386><Z_239><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the sum $$1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 + 88888888 + 999999999$$ and determine what this total is congruent to modulo 9.<|im_end|>
<|im_start|>assistant
<Z_428><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the sum $$1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 + 88888888 + 999999999$$ and determine what this total is congruent to modulo 9.<|im_end|>
<|im_start|>assistant
<Z_184><Z_223><Z_344><Z_320><Z_67><Z_315><Z_209><Z_191><Z_449><Z_235><Z_63><Z_311><Z_125><Z_177><ANSWER>

Question: Calculate the sum $$1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 + 88888888 + 999999999$$ and determine what this total is congruent to modulo 9.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the seven numbers in a list is 60. The average of the first three numbers is 50, and the average of the next two numbers is 70. What is the average of the last two numbers?<|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 of the seven numbers in a list is 60. The average of the first three numbers is 50, and the average of the next two numbers is 70. What is the average of the last two numbers?<|im_end|>
<|im_start|>assistant
<Z_24><Z_356><Z_150><Z_293><Z_240><Z_149><Z_454><Z_116><Z_420><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the seven numbers in a list is 60. The average of the first three numbers is 50, and the average of the next two numbers is 70. What is the average of the last two numbers?<|im_end|>
<|im_start|>assistant
<Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of the seven numbers in a list is 60. The average of the first three numbers is 50, and the average of the next two numbers is 70. What is the average of the last two numbers?<|im_end|>
<|im_start|>assistant
<Z_498><Z_388><Z_483><Z_446><Z_345><Z_235><Z_26><Z_14><Z_480><ANSWER>

Question: The average of the seven numbers in a list is 60. The average of the first three numbers is 50, and the average of the next two numbers is 70. What is the average of the last two numbers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: ${(\sqrt{2} \cdot \sqrt[3]{3})^{6} - \log_{2}(\log_{2}16)}$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: ${(\sqrt{2} \cdot \sqrt[3]{3})^{6} - \log_{2}(\log_{2}16)}$.<|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: ${(\sqrt{2} \cdot \sqrt[3]{3})^{6} - \log_{2}(\log_{2}16)}$.<|im_end|>
<|im_start|>assistant
<Z_290><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: ${(\sqrt{2} \cdot \sqrt[3]{3})^{6} - \log_{2}(\log_{2}16)}$.<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

Question: Compute: ${(\sqrt{2} \cdot \sqrt[3]{3})^{6} - \log_{2}(\log_{2}16)}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many four-digit positive integers have at least one digit that is a 1 or a 7?<|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 four-digit positive integers have at least one digit that is a 1 or a 7?<|im_end|>
<|im_start|>assistant
<Z_369><Z_274><Z_140><Z_224><Z_420><Z_439><Z_443><Z_99><Z_287><Z_103><Z_484><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many four-digit positive integers have at least one digit that is a 1 or a 7?<|im_end|>
<|im_start|>assistant
<Z_15><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many four-digit positive integers have at least one digit that is a 1 or a 7?<|im_end|>
<|im_start|>assistant
<Z_383><Z_305><Z_171><Z_28><Z_365><Z_72><Z_338><Z_226><Z_224><Z_303><Z_453><Z_235><ANSWER>

Question: How many four-digit positive integers have at least one digit that is a 1 or a 7?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jen and Tyler are gymnasts practicing flips. Jen is practicing the triple-flip while Tyler is practicing the double-flip. Jen did sixteen triple-flips during practice. Tyler flipped in the air half the number of times Jen did. How many double-flips did Tyler do?<|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
Jen and Tyler are gymnasts practicing flips. Jen is practicing the triple-flip while Tyler is practicing the double-flip. Jen did sixteen triple-flips during practice. Tyler flipped in the air half the number of times Jen did. How many double-flips did Tyler do?<|im_end|>
<|im_start|>assistant
<Z_477><Z_465><Z_210><Z_419><Z_443><Z_381><Z_343><Z_235><Z_83><Z_323><Z_312><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jen and Tyler are gymnasts practicing flips. Jen is practicing the triple-flip while Tyler is practicing the double-flip. Jen did sixteen triple-flips during practice. Tyler flipped in the air half the number of times Jen did. How many double-flips did Tyler do?<|im_end|>
<|im_start|>assistant
<Z_278><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jen and Tyler are gymnasts practicing flips. Jen is practicing the triple-flip while Tyler is practicing the double-flip. Jen did sixteen triple-flips during practice. Tyler flipped in the air half the number of times Jen did. How many double-flips did Tyler do?<|im_end|>
<|im_start|>assistant
<Z_218><Z_59><Z_336><Z_157><Z_460><Z_439><Z_8><Z_490><Z_267><Z_4><Z_355><Z_171><ANSWER>

Question: Jen and Tyler are gymnasts practicing flips. Jen is practicing the triple-flip while Tyler is practicing the double-flip. Jen did sixteen triple-flips during practice. Tyler flipped in the air half the number of times Jen did. How many double-flips did Tyler do?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Circle \( D \) has a radius of 8 cm. Calculate the area of the largest possible inscribed triangle having one side as a diameter of circle \( D \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Circle \( D \) has a radius of 8 cm. Calculate the area of the largest possible inscribed triangle having one side as a diameter of circle \( D \).<|im_end|>
<|im_start|>assistant
<Z_369><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Circle \( D \) has a radius of 8 cm. Calculate the area of the largest possible inscribed triangle having one side as a diameter of circle \( D \).<|im_end|>
<|im_start|>assistant
<Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Circle \( D \) has a radius of 8 cm. Calculate the area of the largest possible inscribed triangle having one side as a diameter of circle \( D \).<|im_end|>
<|im_start|>assistant
<Z_20><Z_330><ANSWER>

Question: Circle \( D \) has a radius of 8 cm. Calculate the area of the largest possible inscribed triangle having one side as a diameter of circle \( D \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It is said that adding 12 to 60 and then dividing by 3 is equal to a number subtracted by 12 and then multiplied by 3. Find the number.<|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
It is said that adding 12 to 60 and then dividing by 3 is equal to a number subtracted by 12 and then multiplied by 3. Find the number.<|im_end|>
<|im_start|>assistant
<Z_369><Z_117><Z_159><Z_487><Z_503><Z_400><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It is said that adding 12 to 60 and then dividing by 3 is equal to a number subtracted by 12 and then multiplied by 3. Find the number.<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It is said that adding 12 to 60 and then dividing by 3 is equal to a number subtracted by 12 and then multiplied by 3. Find the number.<|im_end|>
<|im_start|>assistant
<Z_443><Z_260><Z_380><Z_456><Z_174><Z_511><Z_253><ANSWER>

Question: It is said that adding 12 to 60 and then dividing by 3 is equal to a number subtracted by 12 and then multiplied by 3. Find the number.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate the expression: some number + sqrt(-4 + 6 × 4 ÷ 3). The result is 13. What is the missing number?<|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
Evaluate the expression: some number + sqrt(-4 + 6 × 4 ÷ 3). The result is 13. What is the missing number?<|im_end|>
<|im_start|>assistant
<Z_109><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate the expression: some number + sqrt(-4 + 6 × 4 ÷ 3). The result is 13. What is the missing number?<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

Question: Evaluate the expression: some number + sqrt(-4 + 6 × 4 ÷ 3). The result is 13. What is the missing number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546.<|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 greatest common divisor (GCD) of the three numbers 390, 455, and 546.<|im_end|>
<|im_start|>assistant
<Z_369><Z_131><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546.<|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
Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546.<|im_end|>
<|im_start|>assistant
<Z_32><Z_110><ANSWER>

Question: Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill is trying to count the toddlers at his daycare, but they keep running around. He double-counts 8 toddlers and misses 3 who are hiding. He thinks he counted a certain number of toddlers, but there are really 21. How many toddlers does Bill think he counted?<|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
Bill is trying to count the toddlers at his daycare, but they keep running around. He double-counts 8 toddlers and misses 3 who are hiding. He thinks he counted a certain number of toddlers, but there are really 21. How many toddlers does Bill think he counted?<|im_end|>
<|im_start|>assistant
<Z_369><Z_125><Z_451><Z_266><Z_406><Z_295><Z_305><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill is trying to count the toddlers at his daycare, but they keep running around. He double-counts 8 toddlers and misses 3 who are hiding. He thinks he counted a certain number of toddlers, but there are really 21. How many toddlers does Bill think he counted?<|im_end|>
<|im_start|>assistant
<Z_384><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill is trying to count the toddlers at his daycare, but they keep running around. He double-counts 8 toddlers and misses 3 who are hiding. He thinks he counted a certain number of toddlers, but there are really 21. How many toddlers does Bill think he counted?<|im_end|>
<|im_start|>assistant
<Z_396><Z_488><Z_487><Z_252><Z_277><Z_469><Z_64><ANSWER>

Question: Bill is trying to count the toddlers at his daycare, but they keep running around. He double-counts 8 toddlers and misses 3 who are hiding. He thinks he counted a certain number of toddlers, but there are really 21. How many toddlers does Bill think he counted?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A forester is planting trees in a forest which already has 30 native trees. On Monday, the forester triples the number of total trees in the forest by planting new trees. On Tuesday, he plants a third of the amount he planted on Monday. On Wednesday, he plants twice the number of trees as he did on Tuesday. If the forester's goal is to achieve a specific total of 300 trees in the forest by the end of Wednesday, how many trees has the forester planted in total?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A forester is planting trees in a forest which already has 30 native trees. On Monday, the forester triples the number of total trees in the forest by planting new trees. On Tuesday, he plants a third of the amount he planted on Monday. On Wednesday, he plants twice the number of trees as he did on Tuesday. If the forester's goal is to achieve a specific total of 300 trees in the forest by the end of Wednesday, how many trees has the forester planted in total?<|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 forester is planting trees in a forest which already has 30 native trees. On Monday, the forester triples the number of total trees in the forest by planting new trees. On Tuesday, he plants a third of the amount he planted on Monday. On Wednesday, he plants twice the number of trees as he did on Tuesday. If the forester's goal is to achieve a specific total of 300 trees in the forest by the end of Wednesday, how many trees has the forester planted in total?<|im_end|>
<|im_start|>assistant
<Z_291><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A forester is planting trees in a forest which already has 30 native trees. On Monday, the forester triples the number of total trees in the forest by planting new trees. On Tuesday, he plants a third of the amount he planted on Monday. On Wednesday, he plants twice the number of trees as he did on Tuesday. If the forester's goal is to achieve a specific total of 300 trees in the forest by the end of Wednesday, how many trees has the forester planted in total?<|im_end|>
<|im_start|>assistant
<Z_507><ANSWER>

Question: A forester is planting trees in a forest which already has 30 native trees. On Monday, the forester triples the number of total trees in the forest by planting new trees. On Tuesday, he plants a third of the amount he planted on Monday. On Wednesday, he plants twice the number of trees as he did on Tuesday. If the forester's goal is to achieve a specific total of 300 trees in the forest by the end of Wednesday, how many trees has the forester planted in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $13$ of her friends, she'll have $3$ cookies left over. If she divides the cookies equally among $8$ of her friends, she'll have $5$ cookies left over. Assuming that Eleanor made fewer than $120$ cookies, what is the sum of the possible numbers of cookies that she could have made?<|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
Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $13$ of her friends, she'll have $3$ cookies left over. If she divides the cookies equally among $8$ of her friends, she'll have $5$ cookies left over. Assuming that Eleanor made fewer than $120$ cookies, what is the sum of the possible numbers of cookies that she could have made?<|im_end|>
<|im_start|>assistant
<Z_24><Z_349><Z_193><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $13$ of her friends, she'll have $3$ cookies left over. If she divides the cookies equally among $8$ of her friends, she'll have $5$ cookies left over. Assuming that Eleanor made fewer than $120$ cookies, what is the sum of the possible numbers of cookies that she could have made?<|im_end|>
<|im_start|>assistant
<Z_434><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $13$ of her friends, she'll have $3$ cookies left over. If she divides the cookies equally among $8$ of her friends, she'll have $5$ cookies left over. Assuming that Eleanor made fewer than $120$ cookies, what is the sum of the possible numbers of cookies that she could have made?<|im_end|>
<|im_start|>assistant
<Z_325><Z_85><Z_128><ANSWER>

Question: Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $13$ of her friends, she'll have $3$ cookies left over. If she divides the cookies equally among $8$ of her friends, she'll have $5$ cookies left over. Assuming that Eleanor made fewer than $120$ cookies, what is the sum of the possible numbers of cookies that she could have made?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Juvy has a garden that has 20 rows with 10 plants in each row. She plants parsley in the first 3 rows, rosemary in some rows, and 150 chives in the rest of the rows. How many rows does Juvy plant rosemary in?<|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
Juvy has a garden that has 20 rows with 10 plants in each row. She plants parsley in the first 3 rows, rosemary in some rows, and 150 chives in the rest of the rows. How many rows does Juvy plant rosemary in?<|im_end|>
<|im_start|>assistant
<Z_24><Z_224><Z_219><Z_127><Z_414><Z_320><Z_178><Z_490><Z_506><Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Juvy has a garden that has 20 rows with 10 plants in each row. She plants parsley in the first 3 rows, rosemary in some rows, and 150 chives in the rest of the rows. How many rows does Juvy plant rosemary in?<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Juvy has a garden that has 20 rows with 10 plants in each row. She plants parsley in the first 3 rows, rosemary in some rows, and 150 chives in the rest of the rows. How many rows does Juvy plant rosemary in?<|im_end|>
<|im_start|>assistant
<Z_147><Z_75><Z_76><Z_478><Z_509><Z_68><Z_504><Z_169><Z_138><Z_341><ANSWER>

Question: Juvy has a garden that has 20 rows with 10 plants in each row. She plants parsley in the first 3 rows, rosemary in some rows, and 150 chives in the rest of the rows. How many rows does Juvy plant rosemary in?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Frank and his friends Betty and Bill went to their farm and picked oranges to eat. Betty picked some oranges and Bill picked 12 oranges. Frank picked three times the number that Betty and Bill picked combined. Frank then planted 2 seeds from each of his oranges into the ground and waited 20 years for each seed to turn into a beautiful orange tree. Each orange tree contains 5 oranges for Frank's son Philip to pick. Philip can pick a total of 810 oranges. How many oranges did Betty pick?<|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
Frank and his friends Betty and Bill went to their farm and picked oranges to eat. Betty picked some oranges and Bill picked 12 oranges. Frank picked three times the number that Betty and Bill picked combined. Frank then planted 2 seeds from each of his oranges into the ground and waited 20 years for each seed to turn into a beautiful orange tree. Each orange tree contains 5 oranges for Frank's son Philip to pick. Philip can pick a total of 810 oranges. How many oranges did Betty pick?<|im_end|>
<|im_start|>assistant
<Z_369><Z_360><Z_340><Z_319><Z_41><Z_273><Z_330><Z_469><Z_271><Z_378><Z_427><Z_214><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Frank and his friends Betty and Bill went to their farm and picked oranges to eat. Betty picked some oranges and Bill picked 12 oranges. Frank picked three times the number that Betty and Bill picked combined. Frank then planted 2 seeds from each of his oranges into the ground and waited 20 years for each seed to turn into a beautiful orange tree. Each orange tree contains 5 oranges for Frank's son Philip to pick. Philip can pick a total of 810 oranges. How many oranges did Betty pick?<|im_end|>
<|im_start|>assistant
<Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Frank and his friends Betty and Bill went to their farm and picked oranges to eat. Betty picked some oranges and Bill picked 12 oranges. Frank picked three times the number that Betty and Bill picked combined. Frank then planted 2 seeds from each of his oranges into the ground and waited 20 years for each seed to turn into a beautiful orange tree. Each orange tree contains 5 oranges for Frank's son Philip to pick. Philip can pick a total of 810 oranges. How many oranges did Betty pick?<|im_end|>
<|im_start|>assistant
<Z_110><Z_456><Z_333><Z_366><Z_433><Z_319><Z_204><Z_508><Z_474><Z_79><Z_435><Z_122><ANSWER>

Question: Frank and his friends Betty and Bill went to their farm and picked oranges to eat. Betty picked some oranges and Bill picked 12 oranges. Frank picked three times the number that Betty and Bill picked combined. Frank then planted 2 seeds from each of his oranges into the ground and waited 20 years for each seed to turn into a beautiful orange tree. Each orange tree contains 5 oranges for Frank's son Philip to pick. Philip can pick a total of 810 oranges. How many oranges did Betty pick?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $n$ terms of a geometric sequence is $48$, and the sum of the first $2n$ terms is $60$. Find the sum of the first $3n$ terms.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $n$ terms of a geometric sequence is $48$, and the sum of the first $2n$ terms is $60$. Find the sum of the first $3n$ terms.<|im_end|>
<|im_start|>assistant
<Z_369><Z_226><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $n$ terms of a geometric sequence is $48$, and the sum of the first $2n$ terms is $60$. Find the sum of the first $3n$ terms.<|im_end|>
<|im_start|>assistant
<Z_462><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $n$ terms of a geometric sequence is $48$, and the sum of the first $2n$ terms is $60$. Find the sum of the first $3n$ terms.<|im_end|>
<|im_start|>assistant
<Z_0><Z_504><Z_453><ANSWER>

Question: The sum of the first $n$ terms of a geometric sequence is $48$, and the sum of the first $2n$ terms is $60$. Find the sum of the first $3n$ terms.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A choir can arrange themselves in rows of 9, 10, or 11 without any member left over. What is the minimum number of choir members?<|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 choir can arrange themselves in rows of 9, 10, or 11 without any member left over. What is the minimum number of choir members?<|im_end|>
<|im_start|>assistant
<Z_24><Z_218><Z_494><Z_418><Z_161><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A choir can arrange themselves in rows of 9, 10, or 11 without any member left over. What is the minimum number of choir members?<|im_end|>
<|im_start|>assistant
<Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A choir can arrange themselves in rows of 9, 10, or 11 without any member left over. What is the minimum number of choir members?<|im_end|>
<|im_start|>assistant
<Z_288><Z_396><Z_487><Z_275><Z_321><ANSWER>

Question: A choir can arrange themselves in rows of 9, 10, or 11 without any member left over. What is the minimum number of choir members?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $$f(x) = \log_2(-x^2 + ax + 3)$$, if $f(1) = 2$, find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $$f(x) = \log_2(-x^2 + ax + 3)$$, if $f(1) = 2$, find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $$f(x) = \log_2(-x^2 + ax + 3)$$, if $f(1) = 2$, find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $$f(x) = \log_2(-x^2 + ax + 3)$$, if $f(1) = 2$, find the value of $a$.<|im_end|>
<|im_start|>assistant
<Z_337><ANSWER>

Question: Given the function $$f(x) = \log_2(-x^2 + ax + 3)$$, if $f(1) = 2$, find the value of $a$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $0 \leqslant x \leqslant 2$, find the maximum and minimum values of the function $y = \left(\frac{1}{4}\right)^{x-1} - 4\left(\frac{1}{2}\right)^{x} + 2$.<|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 $0 \leqslant x \leqslant 2$, find the maximum and minimum values of the function $y = \left(\frac{1}{4}\right)^{x-1} - 4\left(\frac{1}{2}\right)^{x} + 2$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_30><Z_70><Z_179><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $0 \leqslant x \leqslant 2$, find the maximum and minimum values of the function $y = \left(\frac{1}{4}\right)^{x-1} - 4\left(\frac{1}{2}\right)^{x} + 2$.<|im_end|>
<|im_start|>assistant
<Z_243><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $0 \leqslant x \leqslant 2$, find the maximum and minimum values of the function $y = \left(\frac{1}{4}\right)^{x-1} - 4\left(\frac{1}{2}\right)^{x} + 2$.<|im_end|>
<|im_start|>assistant
<Z_88><Z_484><Z_143><Z_298><Z_400><ANSWER>

Question: Given that $0 \leqslant x \leqslant 2$, find the maximum and minimum values of the function $y = \left(\frac{1}{4}\right)^{x-1} - 4\left(\frac{1}{2}\right)^{x} + 2$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many five-digit palindromic numbers have a zero in the hundred's place?<|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 five-digit palindromic numbers have a zero in the hundred's place?<|im_end|>
<|im_start|>assistant
<Z_477><Z_169><Z_450><Z_108><Z_308><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many five-digit palindromic numbers have a zero in the hundred's place?<|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
How many five-digit palindromic numbers have a zero in the hundred's place?<|im_end|>
<|im_start|>assistant
<Z_181><Z_338><Z_212><Z_416><Z_252><ANSWER>

Question: How many five-digit palindromic numbers have a zero in the hundred's place?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A local bus tour sold 65 tickets. Senior citizen tickets cost 10 dollars and regular tickets cost 15 dollars. The total sales were 855 dollars. How many regular tickets were sold?<|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 local bus tour sold 65 tickets. Senior citizen tickets cost 10 dollars and regular tickets cost 15 dollars. The total sales were 855 dollars. How many regular tickets were sold?<|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 local bus tour sold 65 tickets. Senior citizen tickets cost 10 dollars and regular tickets cost 15 dollars. The total sales were 855 dollars. How many regular tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A local bus tour sold 65 tickets. Senior citizen tickets cost 10 dollars and regular tickets cost 15 dollars. The total sales were 855 dollars. How many regular tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_250><ANSWER>

Question: A local bus tour sold 65 tickets. Senior citizen tickets cost 10 dollars and regular tickets cost 15 dollars. The total sales were 855 dollars. How many regular tickets were sold?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?<|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, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?<|im_end|>
<|im_start|>assistant
<Z_369><Z_441><Z_89><Z_306><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?<|im_end|>
<|im_start|>assistant
<Z_438><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?<|im_end|>
<|im_start|>assistant
<Z_42><Z_210><Z_307><Z_478><ANSWER>

Question: A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $\left\lceil |{-52.7}|\right\rceil$.<|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
Evaluate $\left\lceil |{-52.7}|\right\rceil$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_219><Z_126><Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $\left\lceil |{-52.7}|\right\rceil$.<|im_end|>
<|im_start|>assistant
<Z_212><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $\left\lceil |{-52.7}|\right\rceil$.<|im_end|>
<|im_start|>assistant
<Z_502><Z_217><Z_30><Z_79><ANSWER>

Question: Evaluate $\left\lceil |{-52.7}|\right\rceil$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the variance of a set of data is $S^{2}=\frac{1}{5}[\left(6-10\right)^{2}+\left(9-10\right)^{2}+\left(a-10\right)^{2}+\left(11-10\right)^{2}+\left(b-10\right)^{2}]=6.8$, then the value of $a^{2}+b^{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the variance of a set of data is $S^{2}=\frac{1}{5}[\left(6-10\right)^{2}+\left(9-10\right)^{2}+\left(a-10\right)^{2}+\left(11-10\right)^{2}+\left(b-10\right)^{2}]=6.8$, then the value of $a^{2}+b^{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the variance of a set of data is $S^{2}=\frac{1}{5}[\left(6-10\right)^{2}+\left(9-10\right)^{2}+\left(a-10\right)^{2}+\left(11-10\right)^{2}+\left(b-10\right)^{2}]=6.8$, then the value of $a^{2}+b^{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_1><Z_264><Z_173><Z_295><Z_311><Z_487><Z_495><Z_376><Z_502><Z_405><Z_386><Z_358><Z_43><Z_145><Z_507><Z_366><Z_388><ANSWER>

Question: Given that the variance of a set of data is $S^{2}=\frac{1}{5}[\left(6-10\right)^{2}+\left(9-10\right)^{2}+\left(a-10\right)^{2}+\left(11-10\right)^{2}+\left(b-10\right)^{2}]=6.8$, then the value of $a^{2}+b^{2}$ is ____.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A digit is written to the right of the units digit of $864$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?<|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 digit is written to the right of the units digit of $864$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?<|im_end|>
<|im_start|>assistant
<Z_477><Z_418><Z_354><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A digit is written to the right of the units digit of $864$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?<|im_end|>
<|im_start|>assistant
<Z_383><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A digit is written to the right of the units digit of $864$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?<|im_end|>
<|im_start|>assistant
<Z_436><Z_42><Z_323><ANSWER>

Question: A digit is written to the right of the units digit of $864$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(n)=\left(\frac{1+i}{1-i}\right)^{n}+\left(\frac{1-i}{1+i}\right)^{n}\ (n\in N)$, determine the number of subsets of the set $\{x\mid x=f(n)\}$.<|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
Given $f(n)=\left(\frac{1+i}{1-i}\right)^{n}+\left(\frac{1-i}{1+i}\right)^{n}\ (n\in N)$, determine the number of subsets of the set $\{x\mid x=f(n)\}$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_60><Z_262><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(n)=\left(\frac{1+i}{1-i}\right)^{n}+\left(\frac{1-i}{1+i}\right)^{n}\ (n\in N)$, determine the number of subsets of the set $\{x\mid x=f(n)\}$.<|im_end|>
<|im_start|>assistant
<Z_184><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(n)=\left(\frac{1+i}{1-i}\right)^{n}+\left(\frac{1-i}{1+i}\right)^{n}\ (n\in N)$, determine the number of subsets of the set $\{x\mid x=f(n)\}$.<|im_end|>
<|im_start|>assistant
<Z_398><Z_30><Z_38><ANSWER>

Question: Given $f(n)=\left(\frac{1+i}{1-i}\right)^{n}+\left(\frac{1-i}{1+i}\right)^{n}\ (n\in N)$, determine the number of subsets of the set $\{x\mid x=f(n)\}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$5^m = 5 \cdot 25^2 \cdot 125^3$. What is the value of $m$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$5^m = 5 \cdot 25^2 \cdot 125^3$. What is the value of $m$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_357><Z_43><Z_502><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$5^m = 5 \cdot 25^2 \cdot 125^3$. What is the value of $m$?<|im_end|>
<|im_start|>assistant
<Z_203><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$5^m = 5 \cdot 25^2 \cdot 125^3$. What is the value of $m$?<|im_end|>
<|im_start|>assistant
<Z_68><Z_195><Z_497><Z_406><ANSWER>

Question: $5^m = 5 \cdot 25^2 \cdot 125^3$. What is the value of $m$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dana has a quadratic of the form $x^2 + bx + 56$, where $b$ is a specific positive number. Using her knowledge of completing the square, Dana rewrites this quadratic in the form $(x+m)^2 + 20$. What is $b$?<|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
Dana has a quadratic of the form $x^2 + bx + 56$, where $b$ is a specific positive number. Using her knowledge of completing the square, Dana rewrites this quadratic in the form $(x+m)^2 + 20$. What is $b$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_421><Z_144><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dana has a quadratic of the form $x^2 + bx + 56$, where $b$ is a specific positive number. Using her knowledge of completing the square, Dana rewrites this quadratic in the form $(x+m)^2 + 20$. What is $b$?<|im_end|>
<|im_start|>assistant
<Z_44><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dana has a quadratic of the form $x^2 + bx + 56$, where $b$ is a specific positive number. Using her knowledge of completing the square, Dana rewrites this quadratic in the form $(x+m)^2 + 20$. What is $b$?<|im_end|>
<|im_start|>assistant
<Z_2><Z_167><Z_352><ANSWER>

Question: Dana has a quadratic of the form $x^2 + bx + 56$, where $b$ is a specific positive number. Using her knowledge of completing the square, Dana rewrites this quadratic in the form $(x+m)^2 + 20$. What is $b$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \( mn = 23^7 \) and that \( m \) has a units digit of 9, find the units digit of \( n \).<|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
Given that \( mn = 23^7 \) and that \( m \) has a units digit of 9, find the units digit of \( n \).<|im_end|>
<|im_start|>assistant
<Z_477><Z_330><Z_331><Z_111><Z_218><Z_162><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \( mn = 23^7 \) and that \( m \) has a units digit of 9, find the units digit of \( n \).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that \( mn = 23^7 \) and that \( m \) has a units digit of 9, find the units digit of \( n \).<|im_end|>
<|im_start|>assistant
<Z_183><Z_151><Z_88><Z_234><Z_169><Z_168><ANSWER>

Question: Given that \( mn = 23^7 \) and that \( m \) has a units digit of 9, find the units digit of \( n \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that a real number $y$ satisfies \[\sqrt{64-y^2} - \sqrt{36-y^2} = 4.\] What is the value of \[\sqrt{64-y^2} + \sqrt{36-y^2}?\]<|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
Suppose that a real number $y$ satisfies \[\sqrt{64-y^2} - \sqrt{36-y^2} = 4.\] What is the value of \[\sqrt{64-y^2} + \sqrt{36-y^2}?\]<|im_end|>
<|im_start|>assistant
<Z_477><Z_328><Z_328><Z_421><Z_387><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that a real number $y$ satisfies \[\sqrt{64-y^2} - \sqrt{36-y^2} = 4.\] What is the value of \[\sqrt{64-y^2} + \sqrt{36-y^2}?\]<|im_end|>
<|im_start|>assistant
<Z_330><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that a real number $y$ satisfies \[\sqrt{64-y^2} - \sqrt{36-y^2} = 4.\] What is the value of \[\sqrt{64-y^2} + \sqrt{36-y^2}?\]<|im_end|>
<|im_start|>assistant
<Z_396><Z_55><Z_427><Z_130><Z_156><ANSWER>

Question: Suppose that a real number $y$ satisfies \[\sqrt{64-y^2} - \sqrt{36-y^2} = 4.\] What is the value of \[\sqrt{64-y^2} + \sqrt{36-y^2}?\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers \( x \) and \( y \) such that \( x \neq y \) and \( \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \), what is the smallest possible value of \( x + y \)?<|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 positive integers \( x \) and \( y \) such that \( x \neq y \) and \( \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \), what is the smallest possible value of \( x + y \)?<|im_end|>
<|im_start|>assistant
<Z_477><Z_413><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers \( x \) and \( y \) such that \( x \neq y \) and \( \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \), what is the smallest possible value of \( x + y \)?<|im_end|>
<|im_start|>assistant
<Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given positive integers \( x \) and \( y \) such that \( x \neq y \) and \( \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \), what is the smallest possible value of \( x + y \)?<|im_end|>
<|im_start|>assistant
<Z_182><Z_90><ANSWER>

Question: Given positive integers \( x \) and \( y \) such that \( x \neq y \) and \( \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \), what is the smallest possible value of \( x + y \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sydney and Conner are having a three day rock collecting contest. Before they start, Sydney has 837 rocks and Conner has 723. On day one, Sydney collects some rocks and Conner collects 8 times as many. On day two Sydney rests and Conner collects 123. On the third day, Sydney collects twice as many as Conner did on the first day. Conner has to collect 27 rocks on day three to make sure he at least ties Sydney. How many rocks did Sydney collect on day one?<|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
Sydney and Conner are having a three day rock collecting contest. Before they start, Sydney has 837 rocks and Conner has 723. On day one, Sydney collects some rocks and Conner collects 8 times as many. On day two Sydney rests and Conner collects 123. On the third day, Sydney collects twice as many as Conner did on the first day. Conner has to collect 27 rocks on day three to make sure he at least ties Sydney. How many rocks did Sydney collect on day one?<|im_end|>
<|im_start|>assistant
<Z_477><Z_329><Z_124><Z_496><Z_296><Z_407><Z_140><Z_214><Z_144><Z_273><Z_343><Z_501><Z_484><Z_104><Z_345><Z_349><Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sydney and Conner are having a three day rock collecting contest. Before they start, Sydney has 837 rocks and Conner has 723. On day one, Sydney collects some rocks and Conner collects 8 times as many. On day two Sydney rests and Conner collects 123. On the third day, Sydney collects twice as many as Conner did on the first day. Conner has to collect 27 rocks on day three to make sure he at least ties Sydney. How many rocks did Sydney collect on day one?<|im_end|>
<|im_start|>assistant
<Z_499><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sydney and Conner are having a three day rock collecting contest. Before they start, Sydney has 837 rocks and Conner has 723. On day one, Sydney collects some rocks and Conner collects 8 times as many. On day two Sydney rests and Conner collects 123. On the third day, Sydney collects twice as many as Conner did on the first day. Conner has to collect 27 rocks on day three to make sure he at least ties Sydney. How many rocks did Sydney collect on day one?<|im_end|>
<|im_start|>assistant
<Z_397><Z_49><Z_318><Z_328><Z_485><Z_25><Z_405><Z_4><Z_120><Z_448><Z_171><Z_280><Z_12><Z_280><Z_274><Z_392><Z_309><ANSWER>

Question: Sydney and Conner are having a three day rock collecting contest. Before they start, Sydney has 837 rocks and Conner has 723. On day one, Sydney collects some rocks and Conner collects 8 times as many. On day two Sydney rests and Conner collects 123. On the third day, Sydney collects twice as many as Conner did on the first day. Conner has to collect 27 rocks on day three to make sure he at least ties Sydney. How many rocks did Sydney collect on day one?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $F_{1}$ and $F_{2}$ are two foci of the ellipse $C$: $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$, then the area of quadrilateral $PF_{1}QF_{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $F_{1}$ and $F_{2}$ are two foci of the ellipse $C$: $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$, then the area of quadrilateral $PF_{1}QF_{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_477><Z_338><Z_232><Z_386><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $F_{1}$ and $F_{2}$ are two foci of the ellipse $C$: $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$, then the area of quadrilateral $PF_{1}QF_{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $F_{1}$ and $F_{2}$ are two foci of the ellipse $C$: $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$, then the area of quadrilateral $PF_{1}QF_{2}$ is ____.<|im_end|>
<|im_start|>assistant
<Z_210><Z_496><Z_31><Z_319><Z_501><ANSWER>

Question: Given $F_{1}$ and $F_{2}$ are two foci of the ellipse $C$: $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$, then the area of quadrilateral $PF_{1}QF_{2}$ is ____.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(p, q, r, s\) be positive integers such that
\[
\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 10 & 6 \\ -15 & -9 \end{pmatrix}.
\]
Find the smallest possible value of \(p + q + r + s.\)<|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
Let \(p, q, r, s\) be positive integers such that
\[
\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 10 & 6 \\ -15 & -9 \end{pmatrix}.
\]
Find the smallest possible value of \(p + q + r + s.\)<|im_end|>
<|im_start|>assistant
<Z_24><Z_186><Z_331><Z_76><Z_485><Z_439><Z_473><Z_176><Z_318><Z_395><Z_64><Z_194><Z_433><Z_463><Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(p, q, r, s\) be positive integers such that
\[
\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 10 & 6 \\ -15 & -9 \end{pmatrix}.
\]
Find the smallest possible value of \(p + q + r + s.\)<|im_end|>
<|im_start|>assistant
<Z_207><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(p, q, r, s\) be positive integers such that
\[
\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 10 & 6 \\ -15 & -9 \end{pmatrix}.
\]
Find the smallest possible value of \(p + q + r + s.\)<|im_end|>
<|im_start|>assistant
<Z_75><Z_115><Z_425><Z_111><Z_257><Z_189><Z_212><Z_274><Z_175><Z_388><Z_456><Z_367><Z_149><Z_289><Z_198><ANSWER>

Question:
Let \(p, q, r, s\) be positive integers such that
\[
\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 10 & 6 \\ -15 & -9 \end{pmatrix}.
\]
Find the smallest possible value of \(p + q + r + s.\)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Example 1-7, there are 5 characters: $a, b, c, d, e$. Form a string of 6 characters with the following requirements:
1. The 1st and 6th characters must be consonants $b, c, d$.
2. Each string must contain the two vowels $a, e$, and $a$ and $e$ must not be adjacent.
3. Adjacent consonants cannot be the same.

What is the number of such strings?<|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 Example 1-7, there are 5 characters: $a, b, c, d, e$. Form a string of 6 characters with the following requirements:
1. The 1st and 6th characters must be consonants $b, c, d$.
2. Each string must contain the two vowels $a, e$, and $a$ and $e$ must not be adjacent.
3. Adjacent consonants cannot be the same.

What is the number of such strings?<|im_end|>
<|im_start|>assistant
<Z_24><Z_269><Z_247><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Example 1-7, there are 5 characters: $a, b, c, d, e$. Form a string of 6 characters with the following requirements:
1. The 1st and 6th characters must be consonants $b, c, d$.
2. Each string must contain the two vowels $a, e$, and $a$ and $e$ must not be adjacent.
3. Adjacent consonants cannot be the same.

What is the number of such strings?<|im_end|>
<|im_start|>assistant
<Z_102><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Example 1-7, there are 5 characters: $a, b, c, d, e$. Form a string of 6 characters with the following requirements:
1. The 1st and 6th characters must be consonants $b, c, d$.
2. Each string must contain the two vowels $a, e$, and $a$ and $e$ must not be adjacent.
3. Adjacent consonants cannot be the same.

What is the number of such strings?<|im_end|>
<|im_start|>assistant
<Z_369><Z_366><Z_317><ANSWER>

Question:
In Example 1-7, there are 5 characters: $a, b, c, d, e$. Form a string of 6 characters with the following requirements:
1. The 1st and 6th characters must be consonants $b, c, d$.
2. Each string must contain the two vowels $a, e$, and $a$ and $e$ must not be adjacent.
3. Adjacent consonants cannot be the same.

What is the number of such strings?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A machine-shop cutting tool is shaped like a notched circle, as described. The radius of the circle is $\sqrt{72}$ cm, the length of $AB$ is $8$ cm, and that of $BC$ is $6$ cm. The angle $ABC$ is a right angle. Determine the square of the distance (in centimeters) from point $B$ to the center of the circle.<|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 machine-shop cutting tool is shaped like a notched circle, as described. The radius of the circle is $\sqrt{72}$ cm, the length of $AB$ is $8$ cm, and that of $BC$ is $6$ cm. The angle $ABC$ is a right angle. Determine the square of the distance (in centimeters) from point $B$ to the center of the circle.<|im_end|>
<|im_start|>assistant
<Z_369><Z_68><Z_156><Z_269><Z_251><Z_449><Z_507><Z_73><Z_321><Z_378><Z_388><Z_391><Z_468><Z_169><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A machine-shop cutting tool is shaped like a notched circle, as described. The radius of the circle is $\sqrt{72}$ cm, the length of $AB$ is $8$ cm, and that of $BC$ is $6$ cm. The angle $ABC$ is a right angle. Determine the square of the distance (in centimeters) from point $B$ to the center of the circle.<|im_end|>
<|im_start|>assistant
<Z_340><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A machine-shop cutting tool is shaped like a notched circle, as described. The radius of the circle is $\sqrt{72}$ cm, the length of $AB$ is $8$ cm, and that of $BC$ is $6$ cm. The angle $ABC$ is a right angle. Determine the square of the distance (in centimeters) from point $B$ to the center of the circle.<|im_end|>
<|im_start|>assistant
<Z_51><Z_106><Z_453><Z_476><Z_294><Z_148><Z_435><Z_143><Z_421><Z_239><Z_306><Z_330><Z_473><Z_264><ANSWER>

Question: A machine-shop cutting tool is shaped like a notched circle, as described. The radius of the circle is $\sqrt{72}$ cm, the length of $AB$ is $8$ cm, and that of $BC$ is $6$ cm. The angle $ABC$ is a right angle. Determine the square of the distance (in centimeters) from point $B$ to the center of the circle.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common factor of all three-digit palindromes?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common factor of all three-digit palindromes?<|im_end|>
<|im_start|>assistant
<Z_24><Z_174><Z_460><Z_127><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common factor of all three-digit palindromes?<|im_end|>
<|im_start|>assistant
<Z_178><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common factor of all three-digit palindromes?<|im_end|>
<|im_start|>assistant
<Z_157><Z_36><Z_31><Z_53><Z_376><ANSWER>

Question: What is the greatest common factor of all three-digit palindromes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the biotechnology class of 2000, there were 73 graduates. 32 of the graduates found a job, some continued on to their second degree, and 13 did both. If only 9 people didn't do both, how many graduates continued on to their second degree?<|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 the biotechnology class of 2000, there were 73 graduates. 32 of the graduates found a job, some continued on to their second degree, and 13 did both. If only 9 people didn't do both, how many graduates continued on to their second degree?<|im_end|>
<|im_start|>assistant
<Z_24><Z_272><Z_391><Z_487><Z_250><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the biotechnology class of 2000, there were 73 graduates. 32 of the graduates found a job, some continued on to their second degree, and 13 did both. If only 9 people didn't do both, how many graduates continued on to their second degree?<|im_end|>
<|im_start|>assistant
<Z_84><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the biotechnology class of 2000, there were 73 graduates. 32 of the graduates found a job, some continued on to their second degree, and 13 did both. If only 9 people didn't do both, how many graduates continued on to their second degree?<|im_end|>
<|im_start|>assistant
<Z_434><Z_189><Z_340><Z_180><Z_425><ANSWER>

Question: In the biotechnology class of 2000, there were 73 graduates. 32 of the graduates found a job, some continued on to their second degree, and 13 did both. If only 9 people didn't do both, how many graduates continued on to their second degree?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eliza has some siblings. The total height of all the siblings combined is 330 inches. Two of her siblings are both 66 inches tall. Another sibling is 60 inches tall. If Eliza is 2 inches shorter than the last sibling, how many siblings does Eliza 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
Eliza has some siblings. The total height of all the siblings combined is 330 inches. Two of her siblings are both 66 inches tall. Another sibling is 60 inches tall. If Eliza is 2 inches shorter than the last sibling, how many siblings does Eliza have?<|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
Eliza has some siblings. The total height of all the siblings combined is 330 inches. Two of her siblings are both 66 inches tall. Another sibling is 60 inches tall. If Eliza is 2 inches shorter than the last sibling, how many siblings does Eliza have?<|im_end|>
<|im_start|>assistant
<Z_393><ANSWER>

Question: Eliza has some siblings. The total height of all the siblings combined is 330 inches. Two of her siblings are both 66 inches tall. Another sibling is 60 inches tall. If Eliza is 2 inches shorter than the last sibling, how many siblings does Eliza have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence $\\{a_n\\}$ is defined by the formula $a_n=(-1)^{n+1}(2n-1)$. Compute the sum of the first $100$ terms.<|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 sequence $\\{a_n\\}$ is defined by the formula $a_n=(-1)^{n+1}(2n-1)$. Compute the sum of the first $100$ terms.<|im_end|>
<|im_start|>assistant
<Z_477><Z_151><Z_298><Z_344><Z_394><Z_137><Z_71><Z_334><Z_321><Z_403><Z_87><Z_253><Z_501><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence $\\{a_n\\}$ is defined by the formula $a_n=(-1)^{n+1}(2n-1)$. Compute the sum of the first $100$ terms.<|im_end|>
<|im_start|>assistant
<Z_152><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence $\\{a_n\\}$ is defined by the formula $a_n=(-1)^{n+1}(2n-1)$. Compute the sum of the first $100$ terms.<|im_end|>
<|im_start|>assistant
<Z_78><Z_425><Z_509><Z_343><Z_430><Z_233><Z_29><Z_447><Z_45><Z_94><Z_365><Z_236><Z_449><ANSWER>

Question: A sequence $\\{a_n\\}$ is defined by the formula $a_n=(-1)^{n+1}(2n-1)$. Compute the sum of the first $100$ terms.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(x)=2\cos ^{2}x+ \sqrt {3}\sin 2x+a(a\in\mathbb{R},a$ is a constant$)$.
(1) Find the smallest positive period of $f(x)$ when $x\in\mathbb{R}$;
(2) Find the value of $a$ when the maximum value of $f(x)$ is $4$ and $x\in[0, \frac {π}{2}]$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(x)=2\cos ^{2}x+ \sqrt {3}\sin 2x+a(a\in\mathbb{R},a$ is a constant$)$.
(1) Find the smallest positive period of $f(x)$ when $x\in\mathbb{R}$;
(2) Find the value of $a$ when the maximum value of $f(x)$ is $4$ and $x\in[0, \frac {π}{2}]$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_71><Z_377><Z_393><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(x)=2\cos ^{2}x+ \sqrt {3}\sin 2x+a(a\in\mathbb{R},a$ is a constant$)$.
(1) Find the smallest positive period of $f(x)$ when $x\in\mathbb{R}$;
(2) Find the value of $a$ when the maximum value of $f(x)$ is $4$ and $x\in[0, \frac {π}{2}]$.<|im_end|>
<|im_start|>assistant
<Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(x)=2\cos ^{2}x+ \sqrt {3}\sin 2x+a(a\in\mathbb{R},a$ is a constant$)$.
(1) Find the smallest positive period of $f(x)$ when $x\in\mathbb{R}$;
(2) Find the value of $a$ when the maximum value of $f(x)$ is $4$ and $x\in[0, \frac {π}{2}]$.<|im_end|>
<|im_start|>assistant
<Z_448><Z_373><Z_196><Z_73><ANSWER>

Question:
Given $f(x)=2\cos ^{2}x+ \sqrt {3}\sin 2x+a(a\in\mathbb{R},a$ is a constant$)$.
(1) Find the smallest positive period of $f(x)$ when $x\in\mathbb{R}$;
(2) Find the value of $a$ when the maximum value of $f(x)$ is $4$ and $x\in[0, \frac {π}{2}]$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function f(x) = 3^x + 9^x (t ≤ x ≤ t + 1), if the maximum value of f(x) is 12, then the minimum value of f(x) is _______.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function f(x) = 3^x + 9^x (t ≤ x ≤ t + 1), if the maximum value of f(x) is 12, then the minimum value of f(x) is _______.<|im_end|>
<|im_start|>assistant
<Z_24><Z_508><Z_102><Z_161><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function f(x) = 3^x + 9^x (t ≤ x ≤ t + 1), if the maximum value of f(x) is 12, then the minimum value of f(x) is _______.<|im_end|>
<|im_start|>assistant
<Z_306><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function f(x) = 3^x + 9^x (t ≤ x ≤ t + 1), if the maximum value of f(x) is 12, then the minimum value of f(x) is _______.<|im_end|>
<|im_start|>assistant
<Z_278><Z_187><Z_45><Z_40><ANSWER>

Question: Given the function f(x) = 3^x + 9^x (t ≤ x ≤ t + 1), if the maximum value of f(x) is 12, then the minimum value of f(x) is _______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a race of 250 meters, competitor A covers the distance in 40 seconds, competitor B in 50 seconds, and competitor C in 55 seconds. Determine by how much competitor A beats competitors B and C, and find the time difference between B and C at the finish line.<|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 a race of 250 meters, competitor A covers the distance in 40 seconds, competitor B in 50 seconds, and competitor C in 55 seconds. Determine by how much competitor A beats competitors B and C, and find the time difference between B and C at the finish line.<|im_end|>
<|im_start|>assistant
<Z_477><Z_431><Z_319><Z_362><Z_149><Z_248><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a race of 250 meters, competitor A covers the distance in 40 seconds, competitor B in 50 seconds, and competitor C in 55 seconds. Determine by how much competitor A beats competitors B and C, and find the time difference between B and C at the finish line.<|im_end|>
<|im_start|>assistant
<Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a race of 250 meters, competitor A covers the distance in 40 seconds, competitor B in 50 seconds, and competitor C in 55 seconds. Determine by how much competitor A beats competitors B and C, and find the time difference between B and C at the finish line.<|im_end|>
<|im_start|>assistant
<Z_396><Z_410><Z_377><Z_299><Z_263><Z_420><ANSWER>

Question: In a race of 250 meters, competitor A covers the distance in 40 seconds, competitor B in 50 seconds, and competitor C in 55 seconds. Determine by how much competitor A beats competitors B and C, and find the time difference between B and C at the finish line.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base seven sum of the digits of the product of $53_7$ and $343_7$ (which is $7^3$ in decimal), and express the product in base seven.<|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 seven sum of the digits of the product of $53_7$ and $343_7$ (which is $7^3$ in decimal), and express the product in base seven.<|im_end|>
<|im_start|>assistant
<Z_24><Z_100><Z_248><Z_185><Z_498><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base seven sum of the digits of the product of $53_7$ and $343_7$ (which is $7^3$ in decimal), and express the product in base seven.<|im_end|>
<|im_start|>assistant
<Z_241><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the base seven sum of the digits of the product of $53_7$ and $343_7$ (which is $7^3$ in decimal), and express the product in base seven.<|im_end|>
<|im_start|>assistant
<Z_90><Z_465><Z_328><Z_56><Z_370><ANSWER>

Question: Find the base seven sum of the digits of the product of $53_7$ and $343_7$ (which is $7^3$ in decimal), and express the product in base seven.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For $n \geq 1$, the value of the expression
\[(5^{n+1} + 6^{n+2})^2 - (5^{n+1} - 6^{n+2})^2\] is $k \cdot 30^{n+1}$ for some positive integer $k$. What is $k$?<|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
For $n \geq 1$, the value of the expression
\[(5^{n+1} + 6^{n+2})^2 - (5^{n+1} - 6^{n+2})^2\] is $k \cdot 30^{n+1}$ for some positive integer $k$. What is $k$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_149><Z_445><Z_510><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For $n \geq 1$, the value of the expression
\[(5^{n+1} + 6^{n+2})^2 - (5^{n+1} - 6^{n+2})^2\] is $k \cdot 30^{n+1}$ for some positive integer $k$. What is $k$?<|im_end|>
<|im_start|>assistant
<Z_226><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For $n \geq 1$, the value of the expression
\[(5^{n+1} + 6^{n+2})^2 - (5^{n+1} - 6^{n+2})^2\] is $k \cdot 30^{n+1}$ for some positive integer $k$. What is $k$?<|im_end|>
<|im_start|>assistant
<Z_342><Z_42><Z_373><Z_449><ANSWER>

Question:
For $n \geq 1$, the value of the expression
\[(5^{n+1} + 6^{n+2})^2 - (5^{n+1} - 6^{n+2})^2\] is $k \cdot 30^{n+1}$ for some positive integer $k$. What is $k$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subset of the integers $1,2,\cdots,150$ has the property that none of its members is 4 times another. What is the largest number of members such a subset can have?<|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 subset of the integers $1,2,\cdots,150$ has the property that none of its members is 4 times another. What is the largest number of members such a subset can have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_426><Z_52><Z_360><Z_320><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subset of the integers $1,2,\cdots,150$ has the property that none of its members is 4 times another. What is the largest number of members such a subset can have?<|im_end|>
<|im_start|>assistant
<Z_166><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subset of the integers $1,2,\cdots,150$ has the property that none of its members is 4 times another. What is the largest number of members such a subset can have?<|im_end|>
<|im_start|>assistant
<Z_470><Z_396><Z_456><Z_144><Z_52><ANSWER>

Question: A subset of the integers $1,2,\cdots,150$ has the property that none of its members is 4 times another. What is the largest number of members such a subset can have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gwen earned 8 points for each bag of cans she recycled. If she had 4 bags, but didn't recycle 2 of them, how many points would she have earned?<|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
Gwen earned 8 points for each bag of cans she recycled. If she had 4 bags, but didn't recycle 2 of them, how many points would she have earned?<|im_end|>
<|im_start|>assistant
<Z_24><Z_484><Z_214><Z_195><Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gwen earned 8 points for each bag of cans she recycled. If she had 4 bags, but didn't recycle 2 of them, how many points would she have earned?<|im_end|>
<|im_start|>assistant
<Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gwen earned 8 points for each bag of cans she recycled. If she had 4 bags, but didn't recycle 2 of them, how many points would she have earned?<|im_end|>
<|im_start|>assistant
<Z_289><Z_458><Z_410><Z_11><Z_321><ANSWER>

Question: Gwen earned 8 points for each bag of cans she recycled. If she had 4 bags, but didn't recycle 2 of them, how many points would she have earned?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of $x$ if a cube's volume is $7x$ cubic units and its surface area is $x$ square units?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of $x$ if a cube's volume is $7x$ cubic units and its surface area is $x$ square units?<|im_end|>
<|im_start|>assistant
<Z_24><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of $x$ if a cube's volume is $7x$ cubic units and its surface area is $x$ square units?<|im_end|>
<|im_start|>assistant
<Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of $x$ if a cube's volume is $7x$ cubic units and its surface area is $x$ square units?<|im_end|>
<|im_start|>assistant
<Z_345><Z_399><ANSWER>

Question: What is the value of $x$ if a cube's volume is $7x$ cubic units and its surface area is $x$ square units?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of real numbers $x$ that satisfy the equation \[ |x+1| = |x-2| + |x-5| + |x-6| \]<|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
Determine the number of real numbers $x$ that satisfy the equation \[ |x+1| = |x-2| + |x-5| + |x-6| \]<|im_end|>
<|im_start|>assistant
<Z_477><Z_67><Z_209><Z_319><Z_378><Z_344><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 real numbers $x$ that satisfy the equation \[ |x+1| = |x-2| + |x-5| + |x-6| \]<|im_end|>
<|im_start|>assistant
<Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of real numbers $x$ that satisfy the equation \[ |x+1| = |x-2| + |x-5| + |x-6| \]<|im_end|>
<|im_start|>assistant
<Z_31><Z_103><Z_475><Z_293><Z_376><Z_405><ANSWER>

Question: Determine the number of real numbers $x$ that satisfy the equation \[ |x+1| = |x-2| + |x-5| + |x-6| \]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let's say x / (12 p q) is an even prime number, where x is a positive integer, p and q are distinct prime numbers and q is greater than p. If 2p - q = 3, what is the least value of x?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let's say x / (12 p q) is an even prime number, where x is a positive integer, p and q are distinct prime numbers and q is greater than p. If 2p - q = 3, what is the least value of x?<|im_end|>
<|im_start|>assistant
<Z_477><Z_286><Z_92><Z_418><Z_201><Z_393><Z_199><Z_197><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let's say x / (12 p q) is an even prime number, where x is a positive integer, p and q are distinct prime numbers and q is greater than p. If 2p - q = 3, what is the least value of x?<|im_end|>
<|im_start|>assistant
<Z_146><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let's say x / (12 p q) is an even prime number, where x is a positive integer, p and q are distinct prime numbers and q is greater than p. If 2p - q = 3, what is the least value of x?<|im_end|>
<|im_start|>assistant
<Z_465><Z_8><Z_342><Z_443><Z_392><Z_105><Z_504><Z_174><ANSWER>

Question: Let's say x / (12 p q) is an even prime number, where x is a positive integer, p and q are distinct prime numbers and q is greater than p. If 2p - q = 3, what is the least value of x?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The larger interior angle of a parallelogram is 120 ° more than its smaller interior angle. If a third angle in a triangle formed by extending one of the sides is 40 ° more than the smaller interior angle of the parallelogram, what is the measure of the smaller interior angle of the parallelogram?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The larger interior angle of a parallelogram is 120 ° more than its smaller interior angle. If a third angle in a triangle formed by extending one of the sides is 40 ° more than the smaller interior angle of the parallelogram, what is the measure of the smaller interior angle of the parallelogram?<|im_end|>
<|im_start|>assistant
<Z_24><Z_294><Z_146><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The larger interior angle of a parallelogram is 120 ° more than its smaller interior angle. If a third angle in a triangle formed by extending one of the sides is 40 ° more than the smaller interior angle of the parallelogram, what is the measure of the smaller interior angle of the parallelogram?<|im_end|>
<|im_start|>assistant
<Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The larger interior angle of a parallelogram is 120 ° more than its smaller interior angle. If a third angle in a triangle formed by extending one of the sides is 40 ° more than the smaller interior angle of the parallelogram, what is the measure of the smaller interior angle of the parallelogram?<|im_end|>
<|im_start|>assistant
<Z_431><Z_334><Z_60><ANSWER>

Question: The larger interior angle of a parallelogram is 120 ° more than its smaller interior angle. If a third angle in a triangle formed by extending one of the sides is 40 ° more than the smaller interior angle of the parallelogram, what is the measure of the smaller interior angle of the parallelogram?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest number that exactly divides a certain number, 91, and 840. The greatest number that exactly divides these numbers is 7. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest number that exactly divides a certain number, 91, and 840. The greatest number that exactly divides these numbers is 7. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_369><Z_320><Z_487><Z_206><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest number that exactly divides a certain number, 91, and 840. The greatest number that exactly divides these numbers is 7. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest number that exactly divides a certain number, 91, and 840. The greatest number that exactly divides these numbers is 7. What is the first number?<|im_end|>
<|im_start|>assistant
<Z_52><Z_178><Z_245><Z_23><ANSWER>

Question: Find the greatest number that exactly divides a certain number, 91, and 840. The greatest number that exactly divides these numbers is 7. What is the first number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A courtyard is 25 meters long and 16 meters wide and is to be paved with bricks of certain length and 10 cm width. The total number of bricks required is 20,000. What is the length of each brick in centimeters?<|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 courtyard is 25 meters long and 16 meters wide and is to be paved with bricks of certain length and 10 cm width. The total number of bricks required is 20,000. What is the length of each brick in centimeters?<|im_end|>
<|im_start|>assistant
<Z_24><Z_157><Z_172><Z_67><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A courtyard is 25 meters long and 16 meters wide and is to be paved with bricks of certain length and 10 cm width. The total number of bricks required is 20,000. What is the length of each brick in centimeters?<|im_end|>
<|im_start|>assistant
<Z_177><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A courtyard is 25 meters long and 16 meters wide and is to be paved with bricks of certain length and 10 cm width. The total number of bricks required is 20,000. What is the length of each brick in centimeters?<|im_end|>
<|im_start|>assistant
<Z_158><Z_416><Z_71><Z_506><ANSWER>

Question: A courtyard is 25 meters long and 16 meters wide and is to be paved with bricks of certain length and 10 cm width. The total number of bricks required is 20,000. What is the length of each brick in centimeters?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rationalize the denominator of $\frac{2}{3\sqrt{5} + 2\sqrt{11}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?<|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
Rationalize the denominator of $\frac{2}{3\sqrt{5} + 2\sqrt{11}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_421><Z_200><Z_135><Z_201><Z_423><Z_468><Z_386><Z_351><Z_335><Z_421><Z_89><Z_250><Z_215><Z_264><Z_341><Z_37><Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rationalize the denominator of $\frac{2}{3\sqrt{5} + 2\sqrt{11}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?<|im_end|>
<|im_start|>assistant
<Z_396><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rationalize the denominator of $\frac{2}{3\sqrt{5} + 2\sqrt{11}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?<|im_end|>
<|im_start|>assistant
<Z_52><Z_251><Z_247><Z_92><Z_62><Z_308><Z_320><Z_441><Z_298><Z_94><Z_26><Z_319><Z_284><Z_133><Z_29><Z_428><Z_307><Z_262><ANSWER>

Question: Rationalize the denominator of $\frac{2}{3\sqrt{5} + 2\sqrt{11}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

*Proposed by Oleksandr Rybak, Kiev, Ukraine.*<|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
There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

*Proposed by Oleksandr Rybak, Kiev, Ukraine.*<|im_end|>
<|im_start|>assistant
<Z_477><Z_494><Z_414><Z_373><Z_103><Z_498><Z_232><Z_274><Z_68><Z_115><Z_253><Z_230><Z_78><Z_67><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

*Proposed by Oleksandr Rybak, Kiev, Ukraine.*<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

*Proposed by Oleksandr Rybak, Kiev, Ukraine.*<|im_end|>
<|im_start|>assistant
<Z_419><Z_345><Z_419><Z_13><Z_353><Z_154><Z_384><Z_301><Z_211><Z_109><Z_336><Z_31><Z_454><Z_10><ANSWER>

Question:
There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

*Proposed by Oleksandr Rybak, Kiev, Ukraine.*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points \((x, y)\) represented in this table lie on a straight line. The point \((34, s)\) lies on the same line. What is the value of \(s?\)
\[
\begin{array}{c|c}
x & y \\ \hline
2 & 5 \\
6 & 17 \\
10 & 29 \\
\end{array}
\]<|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 points \((x, y)\) represented in this table lie on a straight line. The point \((34, s)\) lies on the same line. What is the value of \(s?\)
\[
\begin{array}{c|c}
x & y \\ \hline
2 & 5 \\
6 & 17 \\
10 & 29 \\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_24><Z_279><Z_293><Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points \((x, y)\) represented in this table lie on a straight line. The point \((34, s)\) lies on the same line. What is the value of \(s?\)
\[
\begin{array}{c|c}
x & y \\ \hline
2 & 5 \\
6 & 17 \\
10 & 29 \\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_21><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points \((x, y)\) represented in this table lie on a straight line. The point \((34, s)\) lies on the same line. What is the value of \(s?\)
\[
\begin{array}{c|c}
x & y \\ \hline
2 & 5 \\
6 & 17 \\
10 & 29 \\
\end{array}
\]<|im_end|>
<|im_start|>assistant
<Z_425><Z_204><Z_459><Z_263><ANSWER>

Question:
The points \((x, y)\) represented in this table lie on a straight line. The point \((34, s)\) lies on the same line. What is the value of \(s?\)
\[
\begin{array}{c|c}
x & y \\ \hline
2 & 5 \\
6 & 17 \\
10 & 29 \\
\end{array}
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?<|im_end|>
<|im_start|>assistant
<Z_24><Z_387><Z_184><Z_243><Z_177><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?<|im_end|>
<|im_start|>assistant
<Z_208><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?<|im_end|>
<|im_start|>assistant
<Z_89><Z_399><Z_160><Z_486><Z_117><ANSWER>

Question: If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $1010001011_{(2)}$ to base 7.<|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
Convert the binary number $1010001011_{(2)}$ to base 7.<|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
Convert the binary number $1010001011_{(2)}$ to base 7.<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $1010001011_{(2)}$ to base 7.<|im_end|>
<|im_start|>assistant
<Z_466><ANSWER>

Question: Convert the binary number $1010001011_{(2)}$ to base 7.