<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .<|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
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .<|im_end|>
<|im_start|>assistant
<Z_24><Z_368><Z_295><Z_139><Z_246><Z_419><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .<|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
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .<|im_end|>
<|im_start|>assistant
<Z_401><Z_251><Z_146><Z_352><Z_191><Z_457><ANSWER>

Question:
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven cards numbered from $1$ to $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either strictly ascending or strictly descending order.<|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
Seven cards numbered from $1$ to $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either strictly ascending or strictly descending order.<|im_end|>
<|im_start|>assistant
<Z_477><Z_443><Z_322><Z_475><Z_361><Z_403><Z_149><Z_138><Z_272><Z_472><Z_471><Z_226><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven cards numbered from $1$ to $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either strictly ascending or strictly descending order.<|im_end|>
<|im_start|>assistant
<Z_334><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven cards numbered from $1$ to $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either strictly ascending or strictly descending order.<|im_end|>
<|im_start|>assistant
<Z_503><Z_410><Z_314><Z_453><Z_402><Z_232><Z_386><Z_473><Z_347><Z_353><Z_138><Z_308><ANSWER>

Question: Seven cards numbered from $1$ to $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either strictly ascending or strictly descending order.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain bag contains 100 balls - 50 white, 20 green, some yellow, 17 red, and 3 purple. If a ball is to be chosen at random, the probability that the ball will be neither red nor purple is 0.8. How many yellow balls are in the bag?<|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 certain bag contains 100 balls - 50 white, 20 green, some yellow, 17 red, and 3 purple. If a ball is to be chosen at random, the probability that the ball will be neither red nor purple is 0.8. How many yellow balls are in the bag?<|im_end|>
<|im_start|>assistant
<Z_369><Z_281><Z_490><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain bag contains 100 balls - 50 white, 20 green, some yellow, 17 red, and 3 purple. If a ball is to be chosen at random, the probability that the ball will be neither red nor purple is 0.8. How many yellow balls are in the bag?<|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
A certain bag contains 100 balls - 50 white, 20 green, some yellow, 17 red, and 3 purple. If a ball is to be chosen at random, the probability that the ball will be neither red nor purple is 0.8. How many yellow balls are in the bag?<|im_end|>
<|im_start|>assistant
<Z_306><Z_405><Z_417><ANSWER>

Question: A certain bag contains 100 balls - 50 white, 20 green, some yellow, 17 red, and 3 purple. If a ball is to be chosen at random, the probability that the ball will be neither red nor purple is 0.8. How many yellow balls are in the bag?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the mean of numbers 28, x, 42, 78, and 104 is 62, what is the mean of 48, 62, 98, 124, and x?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the mean of numbers 28, x, 42, 78, and 104 is 62, what is the mean of 48, 62, 98, 124, and x?<|im_end|>
<|im_start|>assistant
<Z_369><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the mean of numbers 28, x, 42, 78, and 104 is 62, what is the mean of 48, 62, 98, 124, and x?<|im_end|>
<|im_start|>assistant
<Z_247><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the mean of numbers 28, x, 42, 78, and 104 is 62, what is the mean of 48, 62, 98, 124, and x?<|im_end|>
<|im_start|>assistant
<Z_467><Z_177><ANSWER>

Question: If the mean of numbers 28, x, 42, 78, and 104 is 62, what is the mean of 48, 62, 98, 124, and x?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of all positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$<|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 sum of all positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$<|im_end|>
<|im_start|>assistant
<Z_369><Z_509><Z_483><Z_400><Z_217><Z_349><Z_318><Z_71><Z_250><Z_250><Z_102><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of all positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$<|im_end|>
<|im_start|>assistant
<Z_265><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of all positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$<|im_end|>
<|im_start|>assistant
<Z_465><Z_453><Z_67><Z_491><Z_303><Z_420><Z_112><Z_260><Z_288><Z_509><Z_359><ANSWER>

Question: Find the sum of all positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of moles of H2SO4 formed on combining 1 mole of SO2 and 1 mole of H2O2<|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 number of moles of H2SO4 formed on combining 1 mole of SO2 and 1 mole of H2O2<|im_end|>
<|im_start|>assistant
<Z_477><Z_326><Z_391><Z_382><Z_223><Z_204><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of moles of H2SO4 formed on combining 1 mole of SO2 and 1 mole of H2O2<|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
Find the number of moles of H2SO4 formed on combining 1 mole of SO2 and 1 mole of H2O2<|im_end|>
<|im_start|>assistant
<Z_396><Z_331><Z_58><Z_508><Z_487><Z_13><ANSWER>

Question: Find the number of moles of H2SO4 formed on combining 1 mole of SO2 and 1 mole of H2O2


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a triangle has two sides of lengths 8 and 11 units, then how many different integer lengths can the third side be?<|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 a triangle has two sides of lengths 8 and 11 units, then how many different integer lengths can the third side be?<|im_end|>
<|im_start|>assistant
<Z_24><Z_300><Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a triangle has two sides of lengths 8 and 11 units, then how many different integer lengths can the third side be?<|im_end|>
<|im_start|>assistant
<Z_349><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a triangle has two sides of lengths 8 and 11 units, then how many different integer lengths can the third side be?<|im_end|>
<|im_start|>assistant
<Z_440><Z_347><Z_186><ANSWER>

Question: If a triangle has two sides of lengths 8 and 11 units, then how many different integer lengths can the third side be?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve for $x$: $\frac{7}{8} = \frac{n}{96} = \frac{m+n}{112} = \frac{x-m}{144}$.<|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
Solve for $x$: $\frac{7}{8} = \frac{n}{96} = \frac{m+n}{112} = \frac{x-m}{144}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_326><Z_100><Z_164><Z_326><Z_420><Z_54><Z_200><Z_330><Z_43><Z_89><Z_264><Z_146><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve for $x$: $\frac{7}{8} = \frac{n}{96} = \frac{m+n}{112} = \frac{x-m}{144}$.<|im_end|>
<|im_start|>assistant
<Z_304><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve for $x$: $\frac{7}{8} = \frac{n}{96} = \frac{m+n}{112} = \frac{x-m}{144}$.<|im_end|>
<|im_start|>assistant
<Z_392><Z_72><Z_72><Z_387><Z_262><Z_501><Z_117><Z_405><Z_227><Z_467><Z_434><Z_480><Z_90><ANSWER>

Question: Solve for $x$: $\frac{7}{8} = \frac{n}{96} = \frac{m+n}{112} = \frac{x-m}{144}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle \(ABC\), point \(D\) is taken on side \(AC\) such that \(AD = 3\), \(\cos \angle BDC = \frac{13}{20}\), and \(\angle B + \angle ADB = 180^\circ\). Find the perimeter of triangle \(ABC\) if \(BC = 2\).<|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
In triangle \(ABC\), point \(D\) is taken on side \(AC\) such that \(AD = 3\), \(\cos \angle BDC = \frac{13}{20}\), and \(\angle B + \angle ADB = 180^\circ\). Find the perimeter of triangle \(ABC\) if \(BC = 2\).<|im_end|>
<|im_start|>assistant
<Z_24><Z_108><Z_273><Z_482><Z_311><Z_425><Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle \(ABC\), point \(D\) is taken on side \(AC\) such that \(AD = 3\), \(\cos \angle BDC = \frac{13}{20}\), and \(\angle B + \angle ADB = 180^\circ\). Find the perimeter of triangle \(ABC\) if \(BC = 2\).<|im_end|>
<|im_start|>assistant
<Z_474><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle \(ABC\), point \(D\) is taken on side \(AC\) such that \(AD = 3\), \(\cos \angle BDC = \frac{13}{20}\), and \(\angle B + \angle ADB = 180^\circ\). Find the perimeter of triangle \(ABC\) if \(BC = 2\).<|im_end|>
<|im_start|>assistant
<Z_363><Z_428><Z_363><Z_164><Z_1><Z_244><Z_77><ANSWER>

Question: In triangle \(ABC\), point \(D\) is taken on side \(AC\) such that \(AD = 3\), \(\cos \angle BDC = \frac{13}{20}\), and \(\angle B + \angle ADB = 180^\circ\). Find the perimeter of triangle \(ABC\) if \(BC = 2\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Susie and Britney each keep chickens, of two different breeds. Susie has 11 Rhode Island Reds and 6 Golden Comets. Britney has twice as many Rhode Island Reds as Susie, but only half as many Golden Comets. How many more chickens are in Britney's flock than in Susie's?<|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
Susie and Britney each keep chickens, of two different breeds. Susie has 11 Rhode Island Reds and 6 Golden Comets. Britney has twice as many Rhode Island Reds as Susie, but only half as many Golden Comets. How many more chickens are in Britney's flock than in Susie's?<|im_end|>
<|im_start|>assistant
<Z_477><Z_473><Z_211><Z_371><Z_260><Z_341><Z_464><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Susie and Britney each keep chickens, of two different breeds. Susie has 11 Rhode Island Reds and 6 Golden Comets. Britney has twice as many Rhode Island Reds as Susie, but only half as many Golden Comets. How many more chickens are in Britney's flock than in Susie's?<|im_end|>
<|im_start|>assistant
<Z_449><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Susie and Britney each keep chickens, of two different breeds. Susie has 11 Rhode Island Reds and 6 Golden Comets. Britney has twice as many Rhode Island Reds as Susie, but only half as many Golden Comets. How many more chickens are in Britney's flock than in Susie's?<|im_end|>
<|im_start|>assistant
<Z_21><Z_217><Z_175><Z_349><Z_338><Z_335><Z_215><ANSWER>

Question: Susie and Britney each keep chickens, of two different breeds. Susie has 11 Rhode Island Reds and 6 Golden Comets. Britney has twice as many Rhode Island Reds as Susie, but only half as many Golden Comets. How many more chickens are in Britney's flock than in Susie's?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A grocer has sales for 5 consecutive months, and he wants to get an average sale of Rs. 6000. The sales for the other 4 months are Rs. 5660, Rs. 6200, Rs. 6350, and Rs. 6500. In the sixth month, he should have a sale of Rs. 5870. What was the sale in the first month?<|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 grocer has sales for 5 consecutive months, and he wants to get an average sale of Rs. 6000. The sales for the other 4 months are Rs. 5660, Rs. 6200, Rs. 6350, and Rs. 6500. In the sixth month, he should have a sale of Rs. 5870. What was the sale in the first month?<|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 grocer has sales for 5 consecutive months, and he wants to get an average sale of Rs. 6000. The sales for the other 4 months are Rs. 5660, Rs. 6200, Rs. 6350, and Rs. 6500. In the sixth month, he should have a sale of Rs. 5870. What was the sale in the first month?<|im_end|>
<|im_start|>assistant
<Z_263><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A grocer has sales for 5 consecutive months, and he wants to get an average sale of Rs. 6000. The sales for the other 4 months are Rs. 5660, Rs. 6200, Rs. 6350, and Rs. 6500. In the sixth month, he should have a sale of Rs. 5870. What was the sale in the first month?<|im_end|>
<|im_start|>assistant
<Z_501><ANSWER>

Question: A grocer has sales for 5 consecutive months, and he wants to get an average sale of Rs. 6000. The sales for the other 4 months are Rs. 5660, Rs. 6200, Rs. 6350, and Rs. 6500. In the sixth month, he should have a sale of Rs. 5870. What was the sale in the first month?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tony decided to rent a small cottage. The master bedroom and bath totaled 500 sq ft. The 2 guest bedrooms have a certain area each. And the kitchen, guest bath and living area totaled 600 sq ft. Tony spends $3,000 a month on rent, and he is spending $2 per sq ft of house. What is the area of each guest bedroom?<|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
Tony decided to rent a small cottage. The master bedroom and bath totaled 500 sq ft. The 2 guest bedrooms have a certain area each. And the kitchen, guest bath and living area totaled 600 sq ft. Tony spends $3,000 a month on rent, and he is spending $2 per sq ft of house. What is the area of each guest bedroom?<|im_end|>
<|im_start|>assistant
<Z_369><Z_260><Z_501><Z_322><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tony decided to rent a small cottage. The master bedroom and bath totaled 500 sq ft. The 2 guest bedrooms have a certain area each. And the kitchen, guest bath and living area totaled 600 sq ft. Tony spends $3,000 a month on rent, and he is spending $2 per sq ft of house. What is the area of each guest bedroom?<|im_end|>
<|im_start|>assistant
<Z_116><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tony decided to rent a small cottage. The master bedroom and bath totaled 500 sq ft. The 2 guest bedrooms have a certain area each. And the kitchen, guest bath and living area totaled 600 sq ft. Tony spends $3,000 a month on rent, and he is spending $2 per sq ft of house. What is the area of each guest bedroom?<|im_end|>
<|im_start|>assistant
<Z_381><Z_38><Z_210><Z_361><ANSWER>

Question: Tony decided to rent a small cottage. The master bedroom and bath totaled 500 sq ft. The 2 guest bedrooms have a certain area each. And the kitchen, guest bath and living area totaled 600 sq ft. Tony spends $3,000 a month on rent, and he is spending $2 per sq ft of house. What is the area of each guest bedroom?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?<|im_end|>
<|im_start|>assistant
<Z_477><Z_123><Z_322><Z_123><Z_296><Z_471><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?<|im_end|>
<|im_start|>assistant
<Z_455><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?<|im_end|>
<|im_start|>assistant
<Z_385><Z_315><Z_483><Z_485><Z_110><Z_202><ANSWER>

Question: In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A combination lock has a total of nine buttons, comprising eight trigrams and the Taiji symbol. Four buttons need to be pressed in sequence (each button can only be pressed once). If the sequence matches the correct combination, the lock opens; otherwise, it resets to the initial state. It is known that no two opposite trigram buttons (e.g., Qian and Kun, or Brother and Gen) can be pressed consecutively before pressing the Taiji button. Given that there is only one correct sequence, how many tries at most are needed to open the lock?<|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 combination lock has a total of nine buttons, comprising eight trigrams and the Taiji symbol. Four buttons need to be pressed in sequence (each button can only be pressed once). If the sequence matches the correct combination, the lock opens; otherwise, it resets to the initial state. It is known that no two opposite trigram buttons (e.g., Qian and Kun, or Brother and Gen) can be pressed consecutively before pressing the Taiji button. Given that there is only one correct sequence, how many tries at most are needed to open the lock?<|im_end|>
<|im_start|>assistant
<Z_477><Z_74><Z_351><Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A combination lock has a total of nine buttons, comprising eight trigrams and the Taiji symbol. Four buttons need to be pressed in sequence (each button can only be pressed once). If the sequence matches the correct combination, the lock opens; otherwise, it resets to the initial state. It is known that no two opposite trigram buttons (e.g., Qian and Kun, or Brother and Gen) can be pressed consecutively before pressing the Taiji button. Given that there is only one correct sequence, how many tries at most are needed to open the lock?<|im_end|>
<|im_start|>assistant
<Z_179><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A combination lock has a total of nine buttons, comprising eight trigrams and the Taiji symbol. Four buttons need to be pressed in sequence (each button can only be pressed once). If the sequence matches the correct combination, the lock opens; otherwise, it resets to the initial state. It is known that no two opposite trigram buttons (e.g., Qian and Kun, or Brother and Gen) can be pressed consecutively before pressing the Taiji button. Given that there is only one correct sequence, how many tries at most are needed to open the lock?<|im_end|>
<|im_start|>assistant
<Z_139><Z_399><Z_458><Z_167><ANSWER>

Question: A combination lock has a total of nine buttons, comprising eight trigrams and the Taiji symbol. Four buttons need to be pressed in sequence (each button can only be pressed once). If the sequence matches the correct combination, the lock opens; otherwise, it resets to the initial state. It is known that no two opposite trigram buttons (e.g., Qian and Kun, or Brother and Gen) can be pressed consecutively before pressing the Taiji button. Given that there is only one correct sequence, how many tries at most are needed to open the lock?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A four-inch wooden cube is painted blue on four of its six faces, leaving two opposite faces unpainted. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least one face?<|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 four-inch wooden cube is painted blue on four of its six faces, leaving two opposite faces unpainted. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least one face?<|im_end|>
<|im_start|>assistant
<Z_24><Z_227><Z_313><Z_124><Z_200><Z_391><Z_114><Z_70><Z_152><Z_31><Z_310><Z_245><Z_269><Z_111><Z_227><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A four-inch wooden cube is painted blue on four of its six faces, leaving two opposite faces unpainted. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least one face?<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A four-inch wooden cube is painted blue on four of its six faces, leaving two opposite faces unpainted. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least one face?<|im_end|>
<|im_start|>assistant
<Z_7><Z_508><Z_363><Z_411><Z_247><Z_140><Z_321><Z_348><Z_234><Z_499><Z_76><Z_106><Z_176><Z_311><Z_80><Z_247><ANSWER>

Question: A four-inch wooden cube is painted blue on four of its six faces, leaving two opposite faces unpainted. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least one face?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of d, e, and f is 42 kg. If the average weight of d and e is 35 kg and that of e and f is some kg, and the weight of e is 26 kg, what is the average weight of e and f?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of d, e, and f is 42 kg. If the average weight of d and e is 35 kg and that of e and f is some kg, and the weight of e is 26 kg, what is the average weight of e and f?<|im_end|>
<|im_start|>assistant
<Z_477><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of d, e, and f is 42 kg. If the average weight of d and e is 35 kg and that of e and f is some kg, and the weight of e is 26 kg, what is the average weight of e and f?<|im_end|>
<|im_start|>assistant
<Z_454><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average weight of d, e, and f is 42 kg. If the average weight of d and e is 35 kg and that of e and f is some kg, and the weight of e is 26 kg, what is the average weight of e and f?<|im_end|>
<|im_start|>assistant
<Z_338><Z_144><ANSWER>

Question: The average weight of d, e, and f is 42 kg. If the average weight of d and e is 35 kg and that of e and f is some kg, and the weight of e is 26 kg, what is the average weight of e and f?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Train A leaves the station traveling at 30 miles per hour. After some time, Train B leaves the same station traveling in the same direction at 42 miles per hour. Train A was overtaken by Train B 210 miles from the station. How many hours after Train A did Train B leave the station?<|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
Train A leaves the station traveling at 30 miles per hour. After some time, Train B leaves the same station traveling in the same direction at 42 miles per hour. Train A was overtaken by Train B 210 miles from the station. How many hours after Train A did Train B leave the station?<|im_end|>
<|im_start|>assistant
<Z_24><Z_130><Z_494><Z_472><Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Train A leaves the station traveling at 30 miles per hour. After some time, Train B leaves the same station traveling in the same direction at 42 miles per hour. Train A was overtaken by Train B 210 miles from the station. How many hours after Train A did Train B leave the station?<|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
Train A leaves the station traveling at 30 miles per hour. After some time, Train B leaves the same station traveling in the same direction at 42 miles per hour. Train A was overtaken by Train B 210 miles from the station. How many hours after Train A did Train B leave the station?<|im_end|>
<|im_start|>assistant
<Z_220><Z_121><Z_248><Z_112><Z_117><ANSWER>

Question: Train A leaves the station traveling at 30 miles per hour. After some time, Train B leaves the same station traveling in the same direction at 42 miles per hour. Train A was overtaken by Train B 210 miles from the station. How many hours after Train A did Train B leave the station?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tirzah has a lot of purses and handbags; in total she has 26 purses and 24 handbags. Half the purses and 1/4 the handbags are fake. If the rest are authentic, how many purses and handbags in total are genuine?<|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
Tirzah has a lot of purses and handbags; in total she has 26 purses and 24 handbags. Half the purses and 1/4 the handbags are fake. If the rest are authentic, how many purses and handbags in total are genuine?<|im_end|>
<|im_start|>assistant
<Z_369><Z_184><Z_166><Z_149><Z_87><Z_251><Z_111><Z_69><Z_117><Z_395><Z_169><Z_206><Z_418><Z_240><Z_391><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tirzah has a lot of purses and handbags; in total she has 26 purses and 24 handbags. Half the purses and 1/4 the handbags are fake. If the rest are authentic, how many purses and handbags in total are genuine?<|im_end|>
<|im_start|>assistant
<Z_411><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tirzah has a lot of purses and handbags; in total she has 26 purses and 24 handbags. Half the purses and 1/4 the handbags are fake. If the rest are authentic, how many purses and handbags in total are genuine?<|im_end|>
<|im_start|>assistant
<Z_331><Z_44><Z_294><Z_428><Z_128><Z_26><Z_433><Z_39><Z_165><Z_185><Z_367><Z_81><Z_488><Z_465><Z_107><ANSWER>

Question: Tirzah has a lot of purses and handbags; in total she has 26 purses and 24 handbags. Half the purses and 1/4 the handbags are fake. If the rest are authentic, how many purses and handbags in total are genuine?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 66 goats at a farm and twice as many chickens. The number of ducks is half of the total of the goats and chickens, and the number of pigs is a third of the number of ducks. How many more goats are there than pigs?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 66 goats at a farm and twice as many chickens. The number of ducks is half of the total of the goats and chickens, and the number of pigs is a third of the number of ducks. How many more goats are there than pigs?<|im_end|>
<|im_start|>assistant
<Z_477><Z_329><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 66 goats at a farm and twice as many chickens. The number of ducks is half of the total of the goats and chickens, and the number of pigs is a third of the number of ducks. How many more goats are there than pigs?<|im_end|>
<|im_start|>assistant
<Z_189><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 66 goats at a farm and twice as many chickens. The number of ducks is half of the total of the goats and chickens, and the number of pigs is a third of the number of ducks. How many more goats are there than pigs?<|im_end|>
<|im_start|>assistant
<Z_120><Z_405><ANSWER>

Question: There are 66 goats at a farm and twice as many chickens. The number of ducks is half of the total of the goats and chickens, and the number of pigs is a third of the number of ducks. How many more goats are there than pigs?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some number - [ 28 - { 37 - ( 15 - 17 ) } ] = 56, what is the 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
If some number - [ 28 - { 37 - ( 15 - 17 ) } ] = 56, what is the number?<|im_end|>
<|im_start|>assistant
<Z_369><Z_63><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some number - [ 28 - { 37 - ( 15 - 17 ) } ] = 56, what is the number?<|im_end|>
<|im_start|>assistant
<Z_237><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some number - [ 28 - { 37 - ( 15 - 17 ) } ] = 56, what is the number?<|im_end|>
<|im_start|>assistant
<Z_394><Z_201><Z_84><ANSWER>

Question: If some number - [ 28 - { 37 - ( 15 - 17 ) } ] = 56, what is the number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson begins at  $1$  on the number line. At each step, he remains in place with probability  $85\%$  and increases his position on the number line by  $1$  with probability  $15\%$ . Let  $d_n$  be his position on the number line after  $n$  steps, and let  $E_n$  be the expected value of  $\tfrac{1}{d_n}$ . Find the least  $n$  such that  $\tfrac{1}{E_n}
> 2017$ .<|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
Jackson begins at  $1$  on the number line. At each step, he remains in place with probability  $85\%$  and increases his position on the number line by  $1$  with probability  $15\%$ . Let  $d_n$  be his position on the number line after  $n$  steps, and let  $E_n$  be the expected value of  $\tfrac{1}{d_n}$ . Find the least  $n$  such that  $\tfrac{1}{E_n}
> 2017$ .<|im_end|>
<|im_start|>assistant
<Z_369><Z_502><Z_63><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson begins at  $1$  on the number line. At each step, he remains in place with probability  $85\%$  and increases his position on the number line by  $1$  with probability  $15\%$ . Let  $d_n$  be his position on the number line after  $n$  steps, and let  $E_n$  be the expected value of  $\tfrac{1}{d_n}$ . Find the least  $n$  such that  $\tfrac{1}{E_n}
> 2017$ .<|im_end|>
<|im_start|>assistant
<Z_285><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson begins at  $1$  on the number line. At each step, he remains in place with probability  $85\%$  and increases his position on the number line by  $1$  with probability  $15\%$ . Let  $d_n$  be his position on the number line after  $n$  steps, and let  $E_n$  be the expected value of  $\tfrac{1}{d_n}$ . Find the least  $n$  such that  $\tfrac{1}{E_n}
> 2017$ .<|im_end|>
<|im_start|>assistant
<Z_216><Z_115><Z_19><Z_71><ANSWER>

Question:
Jackson begins at  $1$  on the number line. At each step, he remains in place with probability  $85\%$  and increases his position on the number line by  $1$  with probability  $15\%$ . Let  $d_n$  be his position on the number line after  $n$  steps, and let  $E_n$  be the expected value of  $\tfrac{1}{d_n}$ . Find the least  $n$  such that  $\tfrac{1}{E_n}
> 2017$ .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nikola is saving up for a bag of ant food before he can start his ant farm. He wants to have some ants in his farm. Each ant needs 2 ounces of food. Every ounce of ant food costs $.1. Nikola is saving up by raking leaves. He charges $5 to start a job and then he charges by the leaf. Each leaf he rakes costs 1 penny. He finally saved up enough money after he raked 6,000 leaves and completed 4 jobs. How many ants does he want in his farm?<|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
Nikola is saving up for a bag of ant food before he can start his ant farm. He wants to have some ants in his farm. Each ant needs 2 ounces of food. Every ounce of ant food costs $.1. Nikola is saving up by raking leaves. He charges $5 to start a job and then he charges by the leaf. Each leaf he rakes costs 1 penny. He finally saved up enough money after he raked 6,000 leaves and completed 4 jobs. How many ants does he want in his farm?<|im_end|>
<|im_start|>assistant
<Z_24><Z_463><Z_478><Z_246><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nikola is saving up for a bag of ant food before he can start his ant farm. He wants to have some ants in his farm. Each ant needs 2 ounces of food. Every ounce of ant food costs $.1. Nikola is saving up by raking leaves. He charges $5 to start a job and then he charges by the leaf. Each leaf he rakes costs 1 penny. He finally saved up enough money after he raked 6,000 leaves and completed 4 jobs. How many ants does he want in his farm?<|im_end|>
<|im_start|>assistant
<Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nikola is saving up for a bag of ant food before he can start his ant farm. He wants to have some ants in his farm. Each ant needs 2 ounces of food. Every ounce of ant food costs $.1. Nikola is saving up by raking leaves. He charges $5 to start a job and then he charges by the leaf. Each leaf he rakes costs 1 penny. He finally saved up enough money after he raked 6,000 leaves and completed 4 jobs. How many ants does he want in his farm?<|im_end|>
<|im_start|>assistant
<Z_142><Z_316><Z_42><Z_148><ANSWER>

Question: Nikola is saving up for a bag of ant food before he can start his ant farm. He wants to have some ants in his farm. Each ant needs 2 ounces of food. Every ounce of ant food costs $.1. Nikola is saving up by raking leaves. He charges $5 to start a job and then he charges by the leaf. Each leaf he rakes costs 1 penny. He finally saved up enough money after he raked 6,000 leaves and completed 4 jobs. How many ants does he want in his farm?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each given random integer values, and $H = 8$. The value of a word is the sum of its letters' values. If $MATH$ scores 28 points, $TEAM$ scores 34 points, and $MEET$ scores 30 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each given random integer values, and $H = 8$. The value of a word is the sum of its letters' values. If $MATH$ scores 28 points, $TEAM$ scores 34 points, and $MEET$ scores 30 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_413><Z_321><Z_276><Z_320><Z_126><Z_149><Z_364><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each given random integer values, and $H = 8$. The value of a word is the sum of its letters' values. If $MATH$ scores 28 points, $TEAM$ scores 34 points, and $MEET$ scores 30 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_235><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each given random integer values, and $H = 8$. The value of a word is the sum of its letters' values. If $MATH$ scores 28 points, $TEAM$ scores 34 points, and $MEET$ scores 30 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_500><Z_198><Z_2><Z_314><Z_369><Z_282><Z_501><Z_82><ANSWER>

Question: The letters of the alphabet are each given random integer values, and $H = 8$. The value of a word is the sum of its letters' values. If $MATH$ scores 28 points, $TEAM$ scores 34 points, and $MEET$ scores 30 points, what is the value of $A$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
While preparing for a meeting, Bill fills Dixie cups with water from out of a water cooler. The water cooler initially contains some gallons of water, and each Dixie cup holds 6 ounces of water.  If Bill fills one cup of water per each meeting chair, and there are 5 rows of meeting chairs with 10 chairs in each row, then there are 84 ounces of water left in the cooler after all cups have been filled. How many gallons of water did the cooler initially contain?<|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
While preparing for a meeting, Bill fills Dixie cups with water from out of a water cooler. The water cooler initially contains some gallons of water, and each Dixie cup holds 6 ounces of water.  If Bill fills one cup of water per each meeting chair, and there are 5 rows of meeting chairs with 10 chairs in each row, then there are 84 ounces of water left in the cooler after all cups have been filled. How many gallons of water did the cooler initially contain?<|im_end|>
<|im_start|>assistant
<Z_24><Z_437><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
While preparing for a meeting, Bill fills Dixie cups with water from out of a water cooler. The water cooler initially contains some gallons of water, and each Dixie cup holds 6 ounces of water.  If Bill fills one cup of water per each meeting chair, and there are 5 rows of meeting chairs with 10 chairs in each row, then there are 84 ounces of water left in the cooler after all cups have been filled. How many gallons of water did the cooler initially contain?<|im_end|>
<|im_start|>assistant
<Z_249><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
While preparing for a meeting, Bill fills Dixie cups with water from out of a water cooler. The water cooler initially contains some gallons of water, and each Dixie cup holds 6 ounces of water.  If Bill fills one cup of water per each meeting chair, and there are 5 rows of meeting chairs with 10 chairs in each row, then there are 84 ounces of water left in the cooler after all cups have been filled. How many gallons of water did the cooler initially contain?<|im_end|>
<|im_start|>assistant
<Z_250><Z_43><ANSWER>

Question: While preparing for a meeting, Bill fills Dixie cups with water from out of a water cooler. The water cooler initially contains some gallons of water, and each Dixie cup holds 6 ounces of water.  If Bill fills one cup of water per each meeting chair, and there are 5 rows of meeting chairs with 10 chairs in each row, then there are 84 ounces of water left in the cooler after all cups have been filled. How many gallons of water did the cooler initially contain?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A grocery shop has a sale of Rs. 6535, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. The shop must have a sale of Rs. 4891 in the sixth month. What is the average sale the shop wants to achieve?<|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 grocery shop has a sale of Rs. 6535, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. The shop must have a sale of Rs. 4891 in the sixth month. What is the average sale the shop wants to achieve?<|im_end|>
<|im_start|>assistant
<Z_477><Z_274><Z_283><Z_173><Z_351><Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A grocery shop has a sale of Rs. 6535, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. The shop must have a sale of Rs. 4891 in the sixth month. What is the average sale the shop wants to achieve?<|im_end|>
<|im_start|>assistant
<Z_285><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A grocery shop has a sale of Rs. 6535, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. The shop must have a sale of Rs. 4891 in the sixth month. What is the average sale the shop wants to achieve?<|im_end|>
<|im_start|>assistant
<Z_264><Z_231><Z_45><Z_54><Z_287><Z_62><ANSWER>

Question: A grocery shop has a sale of Rs. 6535, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. The shop must have a sale of Rs. 4891 in the sixth month. What is the average sale the shop wants to achieve?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given variables $x$ and $y$ satisfy $|x| + |y| \leq 1$, the maximum value of $x + 2y$ 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 variables $x$ and $y$ satisfy $|x| + |y| \leq 1$, the maximum value of $x + 2y$ is __________.<|im_end|>
<|im_start|>assistant
<Z_369><Z_223><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given variables $x$ and $y$ satisfy $|x| + |y| \leq 1$, the maximum value of $x + 2y$ is __________.<|im_end|>
<|im_start|>assistant
<Z_163><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given variables $x$ and $y$ satisfy $|x| + |y| \leq 1$, the maximum value of $x + 2y$ is __________.<|im_end|>
<|im_start|>assistant
<Z_192><Z_251><ANSWER>

Question: Given variables $x$ and $y$ satisfy $|x| + |y| \leq 1$, the maximum value of $x + 2y$ is __________.


Question:
Assume \( y_1, y_2, \ldots, y_5 \) are real numbers such that
\[
y_1 + 3y_2 + 6y_3 + 10y_4 + 15y_5 = 3, \\
3y_1 + 6y_2 + 10y_3 + 15y_4 + 21y_5 = 20, \\
6y_1 + 10y_2 + 15y_3 + 21y_4 + 28y_5 = 86, \\
10y_1 + 15y_2 + 21y_3 + 28y_4 + 36y_5 = 225.
\]
Find the value of \( 15y_1 + 21y_2 + 28y_3 + 36y_4 + 45y_5 \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2+bx+c = 0$ be a quadratic whose roots are each three more than the roots of $2x^2-4x-8$. What is $c$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2+bx+c = 0$ be a quadratic whose roots are each three more than the roots of $2x^2-4x-8$. What is $c$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_395><Z_253><Z_294><Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2+bx+c = 0$ be a quadratic whose roots are each three more than the roots of $2x^2-4x-8$. What is $c$?<|im_end|>
<|im_start|>assistant
<Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2+bx+c = 0$ be a quadratic whose roots are each three more than the roots of $2x^2-4x-8$. What is $c$?<|im_end|>
<|im_start|>assistant
<Z_416><Z_20><Z_459><Z_107><Z_127><ANSWER>

Question: Let $x^2+bx+c = 0$ be a quadratic whose roots are each three more than the roots of $2x^2-4x-8$. What is $c$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers $x$ satisfying
\[(x^2 - 6x + 5)^{x^2 - 7x + 6} = 1.\]<|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 sum of all real numbers $x$ satisfying
\[(x^2 - 6x + 5)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_24><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers $x$ satisfying
\[(x^2 - 6x + 5)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers $x$ satisfying
\[(x^2 - 6x + 5)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_202><Z_493><ANSWER>

Question:
Determine the sum of all real numbers $x$ satisfying
\[(x^2 - 6x + 5)^{x^2 - 7x + 6} = 1.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn is a certain amount, and the cost of cultivating wheat is 30 dollars. Mr.Johnson has 18,600 dollars available for cultivating these crops. He used all the land and entire budget and planted 200 acres of wheat. What is the cost of cultivating corn per acre?<|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 Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn is a certain amount, and the cost of cultivating wheat is 30 dollars. Mr.Johnson has 18,600 dollars available for cultivating these crops. He used all the land and entire budget and planted 200 acres of wheat. What is the cost of cultivating corn per acre?<|im_end|>
<|im_start|>assistant
<Z_369><Z_192><Z_418><Z_201><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn is a certain amount, and the cost of cultivating wheat is 30 dollars. Mr.Johnson has 18,600 dollars available for cultivating these crops. He used all the land and entire budget and planted 200 acres of wheat. What is the cost of cultivating corn per acre?<|im_end|>
<|im_start|>assistant
<Z_191><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn is a certain amount, and the cost of cultivating wheat is 30 dollars. Mr.Johnson has 18,600 dollars available for cultivating these crops. He used all the land and entire budget and planted 200 acres of wheat. What is the cost of cultivating corn per acre?<|im_end|>
<|im_start|>assistant
<Z_111><Z_104><Z_179><Z_326><Z_150><ANSWER>

Question: The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn is a certain amount, and the cost of cultivating wheat is 30 dollars. Mr.Johnson has 18,600 dollars available for cultivating these crops. He used all the land and entire budget and planted 200 acres of wheat. What is the cost of cultivating corn per acre?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest constant \( C > 0 \) satisfying the following property:

For any 2023-tuple \(\left(a_{1}, a_{2}, \ldots, a_{2023}\right)\) of pairwise distinct, strictly positive real numbers, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$<|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 largest constant \( C > 0 \) satisfying the following property:

For any 2023-tuple \(\left(a_{1}, a_{2}, \ldots, a_{2023}\right)\) of pairwise distinct, strictly positive real numbers, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$<|im_end|>
<|im_start|>assistant
<Z_24><Z_479><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest constant \( C > 0 \) satisfying the following property:

For any 2023-tuple \(\left(a_{1}, a_{2}, \ldots, a_{2023}\right)\) of pairwise distinct, strictly positive real numbers, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$<|im_end|>
<|im_start|>assistant
<Z_224><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest constant \( C > 0 \) satisfying the following property:

For any 2023-tuple \(\left(a_{1}, a_{2}, \ldots, a_{2023}\right)\) of pairwise distinct, strictly positive real numbers, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$<|im_end|>
<|im_start|>assistant
<Z_267><Z_411><ANSWER>

Question:
Determine the largest constant \( C > 0 \) satisfying the following property:

For any 2023-tuple \(\left(a_{1}, a_{2}, \ldots, a_{2023}\right)\) of pairwise distinct, strictly positive real numbers, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$<|im_end|>
<|im_start|>assistant
<Z_477><Z_188><Z_341><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$<|im_end|>
<|im_start|>assistant
<Z_3><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$<|im_end|>
<|im_start|>assistant
<Z_488><Z_172><Z_56><ANSWER>

Question:
Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jake needs to lose some weight to weigh twice as much as his sister. Together they now weigh 278 pounds, and Jake's present weight is 188 pounds. How many pounds does Jake need to lose to weigh twice as much as his sister?<|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
Jake needs to lose some weight to weigh twice as much as his sister. Together they now weigh 278 pounds, and Jake's present weight is 188 pounds. How many pounds does Jake need to lose to weigh twice as much as his sister?<|im_end|>
<|im_start|>assistant
<Z_494><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jake needs to lose some weight to weigh twice as much as his sister. Together they now weigh 278 pounds, and Jake's present weight is 188 pounds. How many pounds does Jake need to lose to weigh twice as much as his sister?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

Question: Jake needs to lose some weight to weigh twice as much as his sister. Together they now weigh 278 pounds, and Jake's present weight is 188 pounds. How many pounds does Jake need to lose to weigh twice as much as his sister?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ben "One Hunna Dolla" Franklin is flying a kite  $KITE$  such that  $IE$  is the perpendicular bisector of  $KT$ . Let  $IE$  meet  $KT$  at  $R$ . The midpoints of  $KI,IT,TE,EK$  are  $A,N,M,D,$  respectively. Given that  $[MAKE]=18,IT=10,[RAIN]=4,$  find  $[DIME]$ .

Note:  $[X]$  denotes the area of the figure  $X$ .<|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
Ben "One Hunna Dolla" Franklin is flying a kite  $KITE$  such that  $IE$  is the perpendicular bisector of  $KT$ . Let  $IE$  meet  $KT$  at  $R$ . The midpoints of  $KI,IT,TE,EK$  are  $A,N,M,D,$  respectively. Given that  $[MAKE]=18,IT=10,[RAIN]=4,$  find  $[DIME]$ .

Note:  $[X]$  denotes the area of the figure  $X$ .<|im_end|>
<|im_start|>assistant
<Z_477><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ben "One Hunna Dolla" Franklin is flying a kite  $KITE$  such that  $IE$  is the perpendicular bisector of  $KT$ . Let  $IE$  meet  $KT$  at  $R$ . The midpoints of  $KI,IT,TE,EK$  are  $A,N,M,D,$  respectively. Given that  $[MAKE]=18,IT=10,[RAIN]=4,$  find  $[DIME]$ .

Note:  $[X]$  denotes the area of the figure  $X$ .<|im_end|>
<|im_start|>assistant
<Z_174><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ben "One Hunna Dolla" Franklin is flying a kite  $KITE$  such that  $IE$  is the perpendicular bisector of  $KT$ . Let  $IE$  meet  $KT$  at  $R$ . The midpoints of  $KI,IT,TE,EK$  are  $A,N,M,D,$  respectively. Given that  $[MAKE]=18,IT=10,[RAIN]=4,$  find  $[DIME]$ .

Note:  $[X]$  denotes the area of the figure  $X$ .<|im_end|>
<|im_start|>assistant
<Z_288><Z_251><ANSWER>

Question:
Ben "One Hunna Dolla" Franklin is flying a kite  $KITE$  such that  $IE$  is the perpendicular bisector of  $KT$ . Let  $IE$  meet  $KT$  at  $R$ . The midpoints of  $KI,IT,TE,EK$  are  $A,N,M,D,$  respectively. Given that  $[MAKE]=18,IT=10,[RAIN]=4,$  find  $[DIME]$ .

Note:  $[X]$  denotes the area of the figure  $X$ .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 300 m long train crosses a platform in 39 sec while it crosses a signal pole in 9 sec. What is the length of the platform?<|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 300 m long train crosses a platform in 39 sec while it crosses a signal pole in 9 sec. What is the length of the platform?<|im_end|>
<|im_start|>assistant
<Z_477><Z_510><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 300 m long train crosses a platform in 39 sec while it crosses a signal pole in 9 sec. What is the length of the platform?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 300 m long train crosses a platform in 39 sec while it crosses a signal pole in 9 sec. What is the length of the platform?<|im_end|>
<|im_start|>assistant
<Z_135><Z_374><ANSWER>

Question: A 300 m long train crosses a platform in 39 sec while it crosses a signal pole in 9 sec. What is the length of the platform?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $\cos^2 45^\circ + \tan 30^\circ \cdot \sin 60^\circ = \ $<|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: $\cos^2 45^\circ + \tan 30^\circ \cdot \sin 60^\circ = \ $<|im_end|>
<|im_start|>assistant
<Z_24><Z_214><Z_123><Z_200><Z_278><Z_59><Z_185><Z_267><Z_139><Z_271><Z_52><Z_144><Z_395><Z_250><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $\cos^2 45^\circ + \tan 30^\circ \cdot \sin 60^\circ = \ $<|im_end|>
<|im_start|>assistant
<Z_280><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $\cos^2 45^\circ + \tan 30^\circ \cdot \sin 60^\circ = \ $<|im_end|>
<|im_start|>assistant
<Z_20><Z_212><Z_175><Z_303><Z_336><Z_265><Z_94><Z_291><Z_425><Z_218><Z_5><Z_254><Z_444><Z_126><ANSWER>

Question: Calculate: $\cos^2 45^\circ + \tan 30^\circ \cdot \sin 60^\circ = \ $


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.<|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 how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.<|im_end|>
<|im_start|>assistant
<Z_369><Z_158><Z_159><Z_102><Z_180><Z_425><Z_255><Z_348><Z_135><Z_71><Z_255><Z_116><Z_147><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.<|im_end|>
<|im_start|>assistant
<Z_201><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.<|im_end|>
<|im_start|>assistant
<Z_61><Z_190><Z_331><Z_107><Z_238><Z_387><Z_360><Z_219><Z_74><Z_388><Z_348><Z_291><Z_510><ANSWER>

Question: Determine how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A women 's age is three years more than twice her sons age . The sum of their ages is 84 . How old is the son ?<|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 women 's age is three years more than twice her sons age . The sum of their ages is 84 . How old is the son ?<|im_end|>
<|im_start|>assistant
<Z_369><Z_437><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A women 's age is three years more than twice her sons age . The sum of their ages is 84 . How old is the son ?<|im_end|>
<|im_start|>assistant
<Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A women 's age is three years more than twice her sons age . The sum of their ages is 84 . How old is the son ?<|im_end|>
<|im_start|>assistant
<Z_32><Z_149><ANSWER>

Question: A women 's age is three years more than twice her sons age . The sum of their ages is 84 . How old is the son ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).

\[
\begin{aligned}
& M_{1}(-3, -1, 1) \\
& M_{2}(-9, 1, -2) \\
& M_{3}(3, -5, 4) \\
& M_{0}(-7, 0, -1)
\end{aligned}
\]<|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 distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).

\[
\begin{aligned}
& M_{1}(-3, -1, 1) \\
& M_{2}(-9, 1, -2) \\
& M_{3}(3, -5, 4) \\
& M_{0}(-7, 0, -1)
\end{aligned}
\]<|im_end|>
<|im_start|>assistant
<Z_477><Z_253><Z_116><Z_164><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).

\[
\begin{aligned}
& M_{1}(-3, -1, 1) \\
& M_{2}(-9, 1, -2) \\
& M_{3}(3, -5, 4) \\
& M_{0}(-7, 0, -1)
\end{aligned}
\]<|im_end|>
<|im_start|>assistant
<Z_412><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).

\[
\begin{aligned}
& M_{1}(-3, -1, 1) \\
& M_{2}(-9, 1, -2) \\
& M_{3}(3, -5, 4) \\
& M_{0}(-7, 0, -1)
\end{aligned}
\]<|im_end|>
<|im_start|>assistant
<Z_320><Z_294><Z_400><Z_99><Z_85><ANSWER>

Question:
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).

\[
\begin{aligned}
& M_{1}(-3, -1, 1) \\
& M_{2}(-9, 1, -2) \\
& M_{3}(3, -5, 4) \\
& M_{0}(-7, 0, -1)
\end{aligned}
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 35 kids in a classroom. 18 kids are wearing socks, 15 are wearing shoes, and 8 are wearing both socks and shoes. How many kids are barefoot?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 35 kids in a classroom. 18 kids are wearing socks, 15 are wearing shoes, and 8 are wearing both socks and shoes. How many kids are barefoot?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 35 kids in a classroom. 18 kids are wearing socks, 15 are wearing shoes, and 8 are wearing both socks and shoes. How many kids are barefoot?<|im_end|>
<|im_start|>assistant
<Z_293><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 35 kids in a classroom. 18 kids are wearing socks, 15 are wearing shoes, and 8 are wearing both socks and shoes. How many kids are barefoot?<|im_end|>
<|im_start|>assistant
<Z_437><ANSWER>

Question: There are 35 kids in a classroom. 18 kids are wearing socks, 15 are wearing shoes, and 8 are wearing both socks and shoes. How many kids are barefoot?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the time it will take for a train that is 120 meter long to pass a bridge of a certain length, if the speed of the train is 40 km/hour. The time it takes is 25.2 seconds. What is the length of the bridge?<|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
Calculate the time it will take for a train that is 120 meter long to pass a bridge of a certain length, if the speed of the train is 40 km/hour. The time it takes is 25.2 seconds. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_24><Z_454><Z_255><Z_175><Z_305><Z_510><Z_328><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the time it will take for a train that is 120 meter long to pass a bridge of a certain length, if the speed of the train is 40 km/hour. The time it takes is 25.2 seconds. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_503><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the time it will take for a train that is 120 meter long to pass a bridge of a certain length, if the speed of the train is 40 km/hour. The time it takes is 25.2 seconds. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_102><Z_39><Z_234><Z_292><Z_164><Z_199><Z_149><Z_202><ANSWER>

Question: Calculate the time it will take for a train that is 120 meter long to pass a bridge of a certain length, if the speed of the train is 40 km/hour. The time it takes is 25.2 seconds. What is the length of the bridge?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
**p1.** The number  $100$  is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.**p2.** Inside an equilateral triangle of side length  $s$  are three mutually tangent circles of radius  $1$ , each one of which is also tangent to two sides of the triangle, as depicted below. Find  $s$ .
![Image](https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png)**p3.** Color a  $4\times 7$  rectangle so that each of its  $28$  unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.**p4.** (a) Show that the  $y$ -intercept of the line through any two distinct points of the graph of  $f(x) = x^2$  is  $-1$  times the product of the  $x$ -coordinates of the two points.
(b) Find all real valued functions with the property that the  $y$ -intercept of the line through any two distinct points of its graph is  $-1$  times the product of the  $x$ -coordinates. Prove that you have found all such functions and that all functions you have found have this property.**p5.** Let  $n$  be a positive integer. We consider sets  $A \subseteq \{1, 2,..., n\}$  with the property that the equation  $x+y=z$  has no solution with  $x\in A$ ,  $y \in A$ ,  $z \in A$ .
(a) Show that there is a set  $A$  as described above that contains  $[(n + l)/2]$  members where  $[x]$  denotes the largest integer less than or equal to  $x$ .
(b) Show that if  $A$  has the property described above, then the number of members of  $A$  is less than or equal to  $[(n + l)/2]$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).<|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
**p1.** The number  $100$  is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.**p2.** Inside an equilateral triangle of side length  $s$  are three mutually tangent circles of radius  $1$ , each one of which is also tangent to two sides of the triangle, as depicted below. Find  $s$ .
![Image](https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png)**p3.** Color a  $4\times 7$  rectangle so that each of its  $28$  unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.**p4.** (a) Show that the  $y$ -intercept of the line through any two distinct points of the graph of  $f(x) = x^2$  is  $-1$  times the product of the  $x$ -coordinates of the two points.
(b) Find all real valued functions with the property that the  $y$ -intercept of the line through any two distinct points of its graph is  $-1$  times the product of the  $x$ -coordinates. Prove that you have found all such functions and that all functions you have found have this property.**p5.** Let  $n$  be a positive integer. We consider sets  $A \subseteq \{1, 2,..., n\}$  with the property that the equation  $x+y=z$  has no solution with  $x\in A$ ,  $y \in A$ ,  $z \in A$ .
(a) Show that there is a set  $A$  as described above that contains  $[(n + l)/2]$  members where  $[x]$  denotes the largest integer less than or equal to  $x$ .
(b) Show that if  $A$  has the property described above, then the number of members of  $A$  is less than or equal to  $[(n + l)/2]$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).<|im_end|>
<|im_start|>assistant
<Z_477><Z_494><Z_276><Z_456><Z_387><Z_331><Z_378><Z_57><Z_487><Z_67><Z_199><Z_340><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
**p1.** The number  $100$  is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.**p2.** Inside an equilateral triangle of side length  $s$  are three mutually tangent circles of radius  $1$ , each one of which is also tangent to two sides of the triangle, as depicted below. Find  $s$ .
![Image](https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png)**p3.** Color a  $4\times 7$  rectangle so that each of its  $28$  unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.**p4.** (a) Show that the  $y$ -intercept of the line through any two distinct points of the graph of  $f(x) = x^2$  is  $-1$  times the product of the  $x$ -coordinates of the two points.
(b) Find all real valued functions with the property that the  $y$ -intercept of the line through any two distinct points of its graph is  $-1$  times the product of the  $x$ -coordinates. Prove that you have found all such functions and that all functions you have found have this property.**p5.** Let  $n$  be a positive integer. We consider sets  $A \subseteq \{1, 2,..., n\}$  with the property that the equation  $x+y=z$  has no solution with  $x\in A$ ,  $y \in A$ ,  $z \in A$ .
(a) Show that there is a set  $A$  as described above that contains  $[(n + l)/2]$  members where  $[x]$  denotes the largest integer less than or equal to  $x$ .
(b) Show that if  $A$  has the property described above, then the number of members of  $A$  is less than or equal to  $[(n + l)/2]$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).<|im_end|>
<|im_start|>assistant
<Z_371><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
**p1.** The number  $100$  is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.**p2.** Inside an equilateral triangle of side length  $s$  are three mutually tangent circles of radius  $1$ , each one of which is also tangent to two sides of the triangle, as depicted below. Find  $s$ .
![Image](https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png)**p3.** Color a  $4\times 7$  rectangle so that each of its  $28$  unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.**p4.** (a) Show that the  $y$ -intercept of the line through any two distinct points of the graph of  $f(x) = x^2$  is  $-1$  times the product of the  $x$ -coordinates of the two points.
(b) Find all real valued functions with the property that the  $y$ -intercept of the line through any two distinct points of its graph is  $-1$  times the product of the  $x$ -coordinates. Prove that you have found all such functions and that all functions you have found have this property.**p5.** Let  $n$  be a positive integer. We consider sets  $A \subseteq \{1, 2,..., n\}$  with the property that the equation  $x+y=z$  has no solution with  $x\in A$ ,  $y \in A$ ,  $z \in A$ .
(a) Show that there is a set  $A$  as described above that contains  $[(n + l)/2]$  members where  $[x]$  denotes the largest integer less than or equal to  $x$ .
(b) Show that if  $A$  has the property described above, then the number of members of  $A$  is less than or equal to  $[(n + l)/2]$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).<|im_end|>
<|im_start|>assistant
<Z_79><Z_300><Z_80><Z_332><Z_283><Z_261><Z_98><Z_321><Z_144><Z_73><Z_400><Z_343><Z_342><ANSWER>

Question:
**p1.** The number  $100$  is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.**p2.** Inside an equilateral triangle of side length  $s$  are three mutually tangent circles of radius  $1$ , each one of which is also tangent to two sides of the triangle, as depicted below. Find  $s$ .
![Image](https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png)**p3.** Color a  $4\times 7$  rectangle so that each of its  $28$  unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.**p4.** (a) Show that the  $y$ -intercept of the line through any two distinct points of the graph of  $f(x) = x^2$  is  $-1$  times the product of the  $x$ -coordinates of the two points.
(b) Find all real valued functions with the property that the  $y$ -intercept of the line through any two distinct points of its graph is  $-1$  times the product of the  $x$ -coordinates. Prove that you have found all such functions and that all functions you have found have this property.**p5.** Let  $n$  be a positive integer. We consider sets  $A \subseteq \{1, 2,..., n\}$  with the property that the equation  $x+y=z$  has no solution with  $x\in A$ ,  $y \in A$ ,  $z \in A$ .
(a) Show that there is a set  $A$  as described above that contains  $[(n + l)/2]$  members where  $[x]$  denotes the largest integer less than or equal to  $x$ .
(b) Show that if  $A$  has the property described above, then the number of members of  $A$  is less than or equal to  $[(n + l)/2]$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of \(a_1, a_2, \dots, a_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \leq i < j \leq 150} a_i a_j.
\]<|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
Each of \(a_1, a_2, \dots, a_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \leq i < j \leq 150} a_i a_j.
\]<|im_end|>
<|im_start|>assistant
<Z_24><Z_47><Z_391><Z_248><Z_484><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of \(a_1, a_2, \dots, a_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \leq i < j \leq 150} a_i a_j.
\]<|im_end|>
<|im_start|>assistant
<Z_62><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each of \(a_1, a_2, \dots, a_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \leq i < j \leq 150} a_i a_j.
\]<|im_end|>
<|im_start|>assistant
<Z_181><Z_461><Z_75><Z_488><Z_282><ANSWER>

Question:
Each of \(a_1, a_2, \dots, a_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \leq i < j \leq 150} a_i a_j.
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|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 student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|im_end|>
<|im_start|>assistant
<Z_369><Z_248><Z_159><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|im_end|>
<|im_start|>assistant
<Z_330><Z_177><Z_294><ANSWER>

Question: A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|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 product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_346><Z_386><Z_352><Z_420><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_361><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_304><Z_511><Z_259><Z_116><Z_491><ANSWER>

Question: What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_369><Z_449><Z_269><Z_450><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_230><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_216><Z_293><Z_488><Z_79><Z_367><ANSWER>

Question: Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of all integers  $n$  not less than  $3$  such that the measure, in degrees, of an interior angle of a regular  $n$ -gon is an integer.

*2016 CCA Math Bonanza Team #3*<|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
Find the sum of all integers  $n$  not less than  $3$  such that the measure, in degrees, of an interior angle of a regular  $n$ -gon is an integer.

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_24><Z_318><Z_273><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of all integers  $n$  not less than  $3$  such that the measure, in degrees, of an interior angle of a regular  $n$ -gon is an integer.

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_111><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of all integers  $n$  not less than  $3$  such that the measure, in degrees, of an interior angle of a regular  $n$ -gon is an integer.

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_395><Z_297><Z_216><Z_206><ANSWER>

Question:
Find the sum of all integers  $n$  not less than  $3$  such that the measure, in degrees, of an interior angle of a regular  $n$ -gon is an integer.

*2016 CCA Math Bonanza Team #3*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_188><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_361><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_20><Z_15><Z_303><ANSWER>

Question: How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|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
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_24><Z_68><Z_331><Z_458><Z_373><Z_352><Z_223><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_459><Z_483><Z_38><Z_75><Z_56><Z_136><Z_145><ANSWER>

Question: an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .