<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .<|im_end|>
<|im_start|>assistant
<Z_24><Z_310><Z_276><Z_489><Z_245><Z_86><Z_172><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .<|im_end|>
<|im_start|>assistant
<Z_250><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .<|im_end|>
<|im_start|>assistant
<Z_436><Z_465><Z_503><Z_256><Z_405><Z_275><Z_259><ANSWER>

Question: A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a smaller square is 8 cm, and the area of a larger square is 64 cm². The squares are arranged such that the top side of the smaller square is collinear with the bottom side of the larger square. What is the distance from point A at the top left corner of the larger square to point B at the bottom right corner of the smaller square?<|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 perimeter of a smaller square is 8 cm, and the area of a larger square is 64 cm². The squares are arranged such that the top side of the smaller square is collinear with the bottom side of the larger square. What is the distance from point A at the top left corner of the larger square to point B at the bottom right corner of the smaller square?<|im_end|>
<|im_start|>assistant
<Z_24><Z_366><Z_147><Z_185><Z_486><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a smaller square is 8 cm, and the area of a larger square is 64 cm². The squares are arranged such that the top side of the smaller square is collinear with the bottom side of the larger square. What is the distance from point A at the top left corner of the larger square to point B at the bottom right corner of the smaller square?<|im_end|>
<|im_start|>assistant
<Z_374><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The perimeter of a smaller square is 8 cm, and the area of a larger square is 64 cm². The squares are arranged such that the top side of the smaller square is collinear with the bottom side of the larger square. What is the distance from point A at the top left corner of the larger square to point B at the bottom right corner of the smaller square?<|im_end|>
<|im_start|>assistant
<Z_53><Z_404><Z_466><Z_205><Z_81><ANSWER>

Question: The perimeter of a smaller square is 8 cm, and the area of a larger square is 64 cm². The squares are arranged such that the top side of the smaller square is collinear with the bottom side of the larger square. What is the distance from point A at the top left corner of the larger square to point B at the bottom right corner of the smaller square?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3 buffaloes eat as much as 4 cows or 2 oxen . at a farm , there are 15 buffaloes , 8 oxen , and 24 cows . the fodder for these cattle is available for 24 days . if 40 more cows and 15 more buffaloes are brought in , how many days will the fodder last ?<|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
3 buffaloes eat as much as 4 cows or 2 oxen . at a farm , there are 15 buffaloes , 8 oxen , and 24 cows . the fodder for these cattle is available for 24 days . if 40 more cows and 15 more buffaloes are brought in , how many days will the fodder last ?<|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
3 buffaloes eat as much as 4 cows or 2 oxen . at a farm , there are 15 buffaloes , 8 oxen , and 24 cows . the fodder for these cattle is available for 24 days . if 40 more cows and 15 more buffaloes are brought in , how many days will the fodder last ?<|im_end|>
<|im_start|>assistant
<Z_306><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3 buffaloes eat as much as 4 cows or 2 oxen . at a farm , there are 15 buffaloes , 8 oxen , and 24 cows . the fodder for these cattle is available for 24 days . if 40 more cows and 15 more buffaloes are brought in , how many days will the fodder last ?<|im_end|>
<|im_start|>assistant
<Z_67><ANSWER>

Question: 3 buffaloes eat as much as 4 cows or 2 oxen . at a farm , there are 15 buffaloes , 8 oxen , and 24 cows . the fodder for these cattle is available for 24 days . if 40 more cows and 15 more buffaloes are brought in , how many days will the fodder last ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits?<|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 smallest positive four-digit number divisible by 9 which has two even and two odd digits?<|im_end|>
<|im_start|>assistant
<Z_369><Z_289><Z_348><Z_497><Z_366><Z_127><Z_494><Z_210><Z_202><Z_346><Z_150><Z_366><Z_209><Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits?<|im_end|>
<|im_start|>assistant
<Z_453><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits?<|im_end|>
<|im_start|>assistant
<Z_55><Z_447><Z_279><Z_44><Z_425><Z_285><Z_377><Z_340><Z_386><Z_0><Z_505><Z_253><Z_12><Z_418><ANSWER>

Question: What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dr. Harry wants to examine Susan's candy consumption in a week, considering she has a daily limit of candies she can consume and the daily prices of the candies are variable. On Tuesday, she buys 3 candies for a total of $2. On Wednesday, she doesn't buy any candies. On Thursday, she buys 5 candies, but the price per candy has increased to $1. On Friday, she purchases 2 candies for a total cost of $1 while the price per candy has decreased. At the end of the week (Sunday night), she sees she has only 4 candies left and her total spending over the week was $9. Considering she doesn't exceed eating more than 3 candies a day, how many candies did she eat during the week?<|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
Dr. Harry wants to examine Susan's candy consumption in a week, considering she has a daily limit of candies she can consume and the daily prices of the candies are variable. On Tuesday, she buys 3 candies for a total of $2. On Wednesday, she doesn't buy any candies. On Thursday, she buys 5 candies, but the price per candy has increased to $1. On Friday, she purchases 2 candies for a total cost of $1 while the price per candy has decreased. At the end of the week (Sunday night), she sees she has only 4 candies left and her total spending over the week was $9. Considering she doesn't exceed eating more than 3 candies a day, how many candies did she eat during the week?<|im_end|>
<|im_start|>assistant
<Z_477><Z_394><Z_34><Z_472><Z_128><Z_396><Z_192><Z_60><Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dr. Harry wants to examine Susan's candy consumption in a week, considering she has a daily limit of candies she can consume and the daily prices of the candies are variable. On Tuesday, she buys 3 candies for a total of $2. On Wednesday, she doesn't buy any candies. On Thursday, she buys 5 candies, but the price per candy has increased to $1. On Friday, she purchases 2 candies for a total cost of $1 while the price per candy has decreased. At the end of the week (Sunday night), she sees she has only 4 candies left and her total spending over the week was $9. Considering she doesn't exceed eating more than 3 candies a day, how many candies did she eat during the week?<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dr. Harry wants to examine Susan's candy consumption in a week, considering she has a daily limit of candies she can consume and the daily prices of the candies are variable. On Tuesday, she buys 3 candies for a total of $2. On Wednesday, she doesn't buy any candies. On Thursday, she buys 5 candies, but the price per candy has increased to $1. On Friday, she purchases 2 candies for a total cost of $1 while the price per candy has decreased. At the end of the week (Sunday night), she sees she has only 4 candies left and her total spending over the week was $9. Considering she doesn't exceed eating more than 3 candies a day, how many candies did she eat during the week?<|im_end|>
<|im_start|>assistant
<Z_433><Z_117><Z_347><Z_479><Z_263><Z_2><Z_54><Z_480><Z_69><ANSWER>

Question: Dr. Harry wants to examine Susan's candy consumption in a week, considering she has a daily limit of candies she can consume and the daily prices of the candies are variable. On Tuesday, she buys 3 candies for a total of $2. On Wednesday, she doesn't buy any candies. On Thursday, she buys 5 candies, but the price per candy has increased to $1. On Friday, she purchases 2 candies for a total cost of $1 while the price per candy has decreased. At the end of the week (Sunday night), she sees she has only 4 candies left and her total spending over the week was $9. Considering she doesn't exceed eating more than 3 candies a day, how many candies did she eat during the week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dima, an amateur, and Fedya, a professional, broke some firewood and boasted to each other about their achievements. Dima exaggerated his work results by 2 times, and Fedya by 7 times. In total, their boasted amount was three times more than the actual amount of firewood broken. Who broke more firewood and by how much?<|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
Dima, an amateur, and Fedya, a professional, broke some firewood and boasted to each other about their achievements. Dima exaggerated his work results by 2 times, and Fedya by 7 times. In total, their boasted amount was three times more than the actual amount of firewood broken. Who broke more firewood and by how much?<|im_end|>
<|im_start|>assistant
<Z_369><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dima, an amateur, and Fedya, a professional, broke some firewood and boasted to each other about their achievements. Dima exaggerated his work results by 2 times, and Fedya by 7 times. In total, their boasted amount was three times more than the actual amount of firewood broken. Who broke more firewood and by how much?<|im_end|>
<|im_start|>assistant
<Z_298><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dima, an amateur, and Fedya, a professional, broke some firewood and boasted to each other about their achievements. Dima exaggerated his work results by 2 times, and Fedya by 7 times. In total, their boasted amount was three times more than the actual amount of firewood broken. Who broke more firewood and by how much?<|im_end|>
<|im_start|>assistant
<Z_438><Z_139><ANSWER>

Question: Dima, an amateur, and Fedya, a professional, broke some firewood and boasted to each other about their achievements. Dima exaggerated his work results by 2 times, and Fedya by 7 times. In total, their boasted amount was three times more than the actual amount of firewood broken. Who broke more firewood and by how much?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?<|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
Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?<|im_end|>
<|im_start|>assistant
<Z_369><Z_506><Z_85><Z_98><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?<|im_end|>
<|im_start|>assistant
<Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?<|im_end|>
<|im_start|>assistant
<Z_284><Z_248><Z_414><Z_133><Z_319><ANSWER>

Question: Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacqueline had some plums, 18 guavas, and 21 apples. She gave 40 fruits to Jane and had 15 fruits left. How many plums did Jacqueline have initially?<|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
Jacqueline had some plums, 18 guavas, and 21 apples. She gave 40 fruits to Jane and had 15 fruits left. How many plums did Jacqueline have initially?<|im_end|>
<|im_start|>assistant
<Z_477><Z_68><Z_54><Z_208><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacqueline had some plums, 18 guavas, and 21 apples. She gave 40 fruits to Jane and had 15 fruits left. How many plums did Jacqueline have initially?<|im_end|>
<|im_start|>assistant
<Z_487><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jacqueline had some plums, 18 guavas, and 21 apples. She gave 40 fruits to Jane and had 15 fruits left. How many plums did Jacqueline have initially?<|im_end|>
<|im_start|>assistant
<Z_405><Z_193><Z_331><Z_279><Z_208><ANSWER>

Question: Jacqueline had some plums, 18 guavas, and 21 apples. She gave 40 fruits to Jane and had 15 fruits left. How many plums did Jacqueline have initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $9\times 1$ board is completely covered by tiles without overlap; each tile may be $1\times 1$, $2\times 1$, or $3\times 1$ in size and lies completely on the board. Each tile is either red, blue, or green. Calculate the number of ways to tile the $9\times 1$ board such that all three colors are used at least once. Let $N$ denote this number, and give the remainder when $N$ is divided by $1000$.<|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 $9\times 1$ board is completely covered by tiles without overlap; each tile may be $1\times 1$, $2\times 1$, or $3\times 1$ in size and lies completely on the board. Each tile is either red, blue, or green. Calculate the number of ways to tile the $9\times 1$ board such that all three colors are used at least once. Let $N$ denote this number, and give the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_472><Z_269><Z_98><Z_386><Z_76><Z_48><Z_116><Z_393><Z_218><Z_104><Z_215><Z_414><Z_309><Z_390><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $9\times 1$ board is completely covered by tiles without overlap; each tile may be $1\times 1$, $2\times 1$, or $3\times 1$ in size and lies completely on the board. Each tile is either red, blue, or green. Calculate the number of ways to tile the $9\times 1$ board such that all three colors are used at least once. Let $N$ denote this number, and give the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_427><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $9\times 1$ board is completely covered by tiles without overlap; each tile may be $1\times 1$, $2\times 1$, or $3\times 1$ in size and lies completely on the board. Each tile is either red, blue, or green. Calculate the number of ways to tile the $9\times 1$ board such that all three colors are used at least once. Let $N$ denote this number, and give the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_107><Z_322><Z_455><Z_367><Z_67><Z_126><Z_236><Z_126><Z_74><Z_187><Z_463><Z_343><Z_384><Z_315><Z_311><ANSWER>

Question: A $9\times 1$ board is completely covered by tiles without overlap; each tile may be $1\times 1$, $2\times 1$, or $3\times 1$ in size and lies completely on the board. Each tile is either red, blue, or green. Calculate the number of ways to tile the $9\times 1$ board such that all three colors are used at least once. Let $N$ denote this number, and give the remainder when $N$ is divided by $1000$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).<|im_end|>
<|im_start|>assistant
<Z_369><Z_467><Z_171><Z_244><Z_188><Z_508><Z_276><Z_70><Z_406><Z_318><Z_200><Z_381><Z_485><Z_335><Z_177><Z_298><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).<|im_end|>
<|im_start|>assistant
<Z_408><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).<|im_end|>
<|im_start|>assistant
<Z_507><Z_128><Z_458><Z_20><Z_84><Z_324><Z_502><Z_157><Z_26><Z_267><Z_445><Z_497><Z_171><Z_4><Z_164><Z_107><ANSWER>

Question: Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two integers are in the ratio of 1 to 4. If 12 is added to the smaller number, the ratio becomes 1 to 1. What is the larger integer?<|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
Two integers are in the ratio of 1 to 4. If 12 is added to the smaller number, the ratio becomes 1 to 1. What is the larger integer?<|im_end|>
<|im_start|>assistant
<Z_24><Z_63><Z_297><Z_194><Z_311><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two integers are in the ratio of 1 to 4. If 12 is added to the smaller number, the ratio becomes 1 to 1. What is the larger integer?<|im_end|>
<|im_start|>assistant
<Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two integers are in the ratio of 1 to 4. If 12 is added to the smaller number, the ratio becomes 1 to 1. What is the larger integer?<|im_end|>
<|im_start|>assistant
<Z_476><Z_287><Z_332><Z_176><Z_260><ANSWER>

Question: Two integers are in the ratio of 1 to 4. If 12 is added to the smaller number, the ratio becomes 1 to 1. What is the larger integer?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance in 276 min, if it runs at a certain speed on an average. To reduce the time of journey to 69 min, the train must run at a speed of 16 kmph. What is the average speed of the train when it takes 276 min to cover the distance?<|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 train covers a distance in 276 min, if it runs at a certain speed on an average. To reduce the time of journey to 69 min, the train must run at a speed of 16 kmph. What is the average speed of the train when it takes 276 min to cover the distance?<|im_end|>
<|im_start|>assistant
<Z_369><Z_255><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance in 276 min, if it runs at a certain speed on an average. To reduce the time of journey to 69 min, the train must run at a speed of 16 kmph. What is the average speed of the train when it takes 276 min to cover the distance?<|im_end|>
<|im_start|>assistant
<Z_496><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance in 276 min, if it runs at a certain speed on an average. To reduce the time of journey to 69 min, the train must run at a speed of 16 kmph. What is the average speed of the train when it takes 276 min to cover the distance?<|im_end|>
<|im_start|>assistant
<Z_52><Z_19><ANSWER>

Question: A train covers a distance in 276 min, if it runs at a certain speed on an average. To reduce the time of journey to 69 min, the train must run at a speed of 16 kmph. What is the average speed of the train when it takes 276 min to cover the distance?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?<|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
Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?<|im_end|>
<|im_start|>assistant
<Z_369><Z_45><Z_362><Z_431><Z_274><Z_84><Z_318><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?<|im_end|>
<|im_start|>assistant
<Z_136><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?<|im_end|>
<|im_start|>assistant
<Z_489><Z_361><Z_408><Z_408><Z_161><Z_244><Z_456><Z_454><ANSWER>

Question: Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 240 third grade students at Heesu's school. If the number of third grade students who were glasses is 3/5 of the number of students who do not were glasses, how may third grade students do not were glasses?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 240 third grade students at Heesu's school. If the number of third grade students who were glasses is 3/5 of the number of students who do not were glasses, how may third grade students do not were glasses?<|im_end|>
<|im_start|>assistant
<Z_31><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 240 third grade students at Heesu's school. If the number of third grade students who were glasses is 3/5 of the number of students who do not were glasses, how may third grade students do not were glasses?<|im_end|>
<|im_start|>assistant
<Z_101><ANSWER>

Question: There are 240 third grade students at Heesu's school. If the number of third grade students who were glasses is 3/5 of the number of students who do not were glasses, how may third grade students do not were glasses?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three-fifths of the people in a room are seated in four-fifths of the chairs. The rest of the people are standing. If there are 5 empty chairs, how many people are in the room?<|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
Three-fifths of the people in a room are seated in four-fifths of the chairs. The rest of the people are standing. If there are 5 empty chairs, how many people are in the room?<|im_end|>
<|im_start|>assistant
<Z_419><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three-fifths of the people in a room are seated in four-fifths of the chairs. The rest of the people are standing. If there are 5 empty chairs, how many people are in the room?<|im_end|>
<|im_start|>assistant
<Z_167><ANSWER>

Question: Three-fifths of the people in a room are seated in four-fifths of the chairs. The rest of the people are standing. If there are 5 empty chairs, how many people are in the room?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Taking the origin O of the Cartesian coordinate system as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve C is $\rho=1$, and the parametric equation of line $l$ is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=2+ \frac { \sqrt {3}}{2}t \end{cases}$$ (where $t$ is the parameter).
(1) Find the Cartesian equations of line $l$ and curve $C$;
(2) Suppose curve $C$ undergoes a scaling transformation $$\begin{cases} x'=2x \\ y'=y \end{cases}$$ to obtain curve $C'$, and let any point on curve $C'$ be $M(x, y)$. Find the maximum value of $$x+2 \sqrt {3}y$$.<|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
Taking the origin O of the Cartesian coordinate system as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve C is $\rho=1$, and the parametric equation of line $l$ is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=2+ \frac { \sqrt {3}}{2}t \end{cases}$$ (where $t$ is the parameter).
(1) Find the Cartesian equations of line $l$ and curve $C$;
(2) Suppose curve $C$ undergoes a scaling transformation $$\begin{cases} x'=2x \\ y'=y \end{cases}$$ to obtain curve $C'$, and let any point on curve $C'$ be $M(x, y)$. Find the maximum value of $$x+2 \sqrt {3}y$$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_276><Z_223><Z_232><Z_43><Z_485><Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Taking the origin O of the Cartesian coordinate system as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve C is $\rho=1$, and the parametric equation of line $l$ is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=2+ \frac { \sqrt {3}}{2}t \end{cases}$$ (where $t$ is the parameter).
(1) Find the Cartesian equations of line $l$ and curve $C$;
(2) Suppose curve $C$ undergoes a scaling transformation $$\begin{cases} x'=2x \\ y'=y \end{cases}$$ to obtain curve $C'$, and let any point on curve $C'$ be $M(x, y)$. Find the maximum value of $$x+2 \sqrt {3}y$$.<|im_end|>
<|im_start|>assistant
<Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Taking the origin O of the Cartesian coordinate system as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve C is $\rho=1$, and the parametric equation of line $l$ is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=2+ \frac { \sqrt {3}}{2}t \end{cases}$$ (where $t$ is the parameter).
(1) Find the Cartesian equations of line $l$ and curve $C$;
(2) Suppose curve $C$ undergoes a scaling transformation $$\begin{cases} x'=2x \\ y'=y \end{cases}$$ to obtain curve $C'$, and let any point on curve $C'$ be $M(x, y)$. Find the maximum value of $$x+2 \sqrt {3}y$$.<|im_end|>
<|im_start|>assistant
<Z_403><Z_136><Z_339><Z_150><Z_158><Z_170><Z_322><ANSWER>

Question:
Taking the origin O of the Cartesian coordinate system as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve C is $\rho=1$, and the parametric equation of line $l$ is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=2+ \frac { \sqrt {3}}{2}t \end{cases}$$ (where $t$ is the parameter).
(1) Find the Cartesian equations of line $l$ and curve $C$;
(2) Suppose curve $C$ undergoes a scaling transformation $$\begin{cases} x'=2x \\ y'=y \end{cases}$$ to obtain curve $C'$, and let any point on curve $C'$ be $M(x, y)$. Find the maximum value of $$x+2 \sqrt {3}y$$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boy takes 64 seconds to run round a square field of a certain side length, if he runs at the rate of 9 km/hr. What is the side length of the square field?<|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 boy takes 64 seconds to run round a square field of a certain side length, if he runs at the rate of 9 km/hr. What is the side length of the square field?<|im_end|>
<|im_start|>assistant
<Z_24><Z_392><Z_404><Z_451><Z_143><Z_243><Z_450><Z_310><Z_195><Z_470><Z_84><Z_57><Z_278><Z_321><Z_286><Z_224><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boy takes 64 seconds to run round a square field of a certain side length, if he runs at the rate of 9 km/hr. What is the side length of the square field?<|im_end|>
<|im_start|>assistant
<Z_394><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boy takes 64 seconds to run round a square field of a certain side length, if he runs at the rate of 9 km/hr. What is the side length of the square field?<|im_end|>
<|im_start|>assistant
<Z_161><Z_414><Z_503><Z_197><Z_86><Z_315><Z_70><Z_499><Z_151><Z_360><Z_421><Z_228><Z_330><Z_81><Z_101><Z_26><ANSWER>

Question: A boy takes 64 seconds to run round a square field of a certain side length, if he runs at the rate of 9 km/hr. What is the side length of the square field?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A = (0,0)$ and $B = (b,4)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 150^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,4,8,12,16,20\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers, and n is not divisible by the square of any prime. Find $m + n.$<|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 $A = (0,0)$ and $B = (b,4)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 150^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,4,8,12,16,20\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers, and n is not divisible by the square of any prime. Find $m + n.$<|im_end|>
<|im_start|>assistant
<Z_24><Z_471><Z_359><Z_236><Z_177><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A = (0,0)$ and $B = (b,4)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 150^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,4,8,12,16,20\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers, and n is not divisible by the square of any prime. Find $m + n.$<|im_end|>
<|im_start|>assistant
<Z_367><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $A = (0,0)$ and $B = (b,4)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 150^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,4,8,12,16,20\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers, and n is not divisible by the square of any prime. Find $m + n.$<|im_end|>
<|im_start|>assistant
<Z_184><Z_91><Z_245><Z_257><Z_9><Z_182><ANSWER>

Question: Let $A = (0,0)$ and $B = (b,4)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 150^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,4,8,12,16,20\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers, and n is not divisible by the square of any prime. Find $m + n.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A classroom contains 48 chairs, and they need to be arranged in a rectangular array with at least two chairs in every row and column. How many distinct arrays can be formed?<|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 classroom contains 48 chairs, and they need to be arranged in a rectangular array with at least two chairs in every row and column. How many distinct arrays can be formed?<|im_end|>
<|im_start|>assistant
<Z_369><Z_437><Z_510><Z_289><Z_503><Z_349><Z_194><Z_433><Z_267><Z_129><Z_320><Z_173><Z_274><Z_437><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A classroom contains 48 chairs, and they need to be arranged in a rectangular array with at least two chairs in every row and column. How many distinct arrays can be formed?<|im_end|>
<|im_start|>assistant
<Z_184><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A classroom contains 48 chairs, and they need to be arranged in a rectangular array with at least two chairs in every row and column. How many distinct arrays can be formed?<|im_end|>
<|im_start|>assistant
<Z_243><Z_344><Z_154><Z_20><Z_466><Z_287><Z_442><Z_170><Z_36><Z_67><Z_510><Z_373><Z_143><Z_107><ANSWER>

Question: A classroom contains 48 chairs, and they need to be arranged in a rectangular array with at least two chairs in every row and column. How many distinct arrays can be formed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 660 = 0$ has integral solutions?<|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 smallest positive value of $m$ so that the equation $10x^2 - mx + 660 = 0$ has integral solutions?<|im_end|>
<|im_start|>assistant
<Z_477><Z_296><Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 660 = 0$ has integral solutions?<|im_end|>
<|im_start|>assistant
<Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 660 = 0$ has integral solutions?<|im_end|>
<|im_start|>assistant
<Z_246><Z_358><Z_448><ANSWER>

Question: What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 660 = 0$ has integral solutions?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A stick 6 cm long, a stick 10 cm long, and a third stick $m$ cm long form a triangle. What is the sum of all possible whole number values of $m$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A stick 6 cm long, a stick 10 cm long, and a third stick $m$ cm long form a triangle. What is the sum of all possible whole number values of $m$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_312><Z_79><Z_312><Z_174><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A stick 6 cm long, a stick 10 cm long, and a third stick $m$ cm long form a triangle. What is the sum of all possible whole number values of $m$?<|im_end|>
<|im_start|>assistant
<Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A stick 6 cm long, a stick 10 cm long, and a third stick $m$ cm long form a triangle. What is the sum of all possible whole number values of $m$?<|im_end|>
<|im_start|>assistant
<Z_323><Z_218><Z_350><Z_134><Z_236><ANSWER>

Question: A stick 6 cm long, a stick 10 cm long, and a third stick $m$ cm long form a triangle. What is the sum of all possible whole number values of $m$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Blake wants to spend his Friday making milkshakes for his family. He knows that he needs 4 ounces of milk and 12 ounces of ice cream for each milkshake. If he has 72 ounces of milk and 192 ounces of ice cream, how much milk will be left over when he is done?<|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
Blake wants to spend his Friday making milkshakes for his family. He knows that he needs 4 ounces of milk and 12 ounces of ice cream for each milkshake. If he has 72 ounces of milk and 192 ounces of ice cream, how much milk will be left over when he is done?<|im_end|>
<|im_start|>assistant
<Z_477><Z_285><Z_406><Z_175><Z_75><Z_106><Z_425><Z_319><Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Blake wants to spend his Friday making milkshakes for his family. He knows that he needs 4 ounces of milk and 12 ounces of ice cream for each milkshake. If he has 72 ounces of milk and 192 ounces of ice cream, how much milk will be left over when he is done?<|im_end|>
<|im_start|>assistant
<Z_89><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Blake wants to spend his Friday making milkshakes for his family. He knows that he needs 4 ounces of milk and 12 ounces of ice cream for each milkshake. If he has 72 ounces of milk and 192 ounces of ice cream, how much milk will be left over when he is done?<|im_end|>
<|im_start|>assistant
<Z_157><Z_313><Z_262><Z_71><Z_483><Z_158><Z_477><Z_10><Z_371><ANSWER>

Question: Blake wants to spend his Friday making milkshakes for his family. He knows that he needs 4 ounces of milk and 12 ounces of ice cream for each milkshake. If he has 72 ounces of milk and 192 ounces of ice cream, how much milk will be left over when he is done?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
United Telephone charges a base rate of $7.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?<|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
United Telephone charges a base rate of $7.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?<|im_end|>
<|im_start|>assistant
<Z_477><Z_308><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
United Telephone charges a base rate of $7.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?<|im_end|>
<|im_start|>assistant
<Z_323><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
United Telephone charges a base rate of $7.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?<|im_end|>
<|im_start|>assistant
<Z_497><Z_78><Z_257><ANSWER>

Question: United Telephone charges a base rate of $7.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$2\lg 2+ \frac {\lg 5}{\lg \sqrt {10}}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$2\lg 2+ \frac {\lg 5}{\lg \sqrt {10}}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_24><Z_53><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$2\lg 2+ \frac {\lg 5}{\lg \sqrt {10}}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_382><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$2\lg 2+ \frac {\lg 5}{\lg \sqrt {10}}=$ ______.<|im_end|>
<|im_start|>assistant
<Z_123><Z_113><Z_243><ANSWER>

Question: $2\lg 2+ \frac {\lg 5}{\lg \sqrt {10}}=$ ______.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a train 175 m long running at 36 kmph crosses a platform in 40 sec . what is the length of the platform ?<|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 train 175 m long running at 36 kmph crosses a platform in 40 sec . what is the length of the platform ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_302><Z_381><Z_333><Z_323><Z_59><Z_273><Z_135><Z_441><Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a train 175 m long running at 36 kmph crosses a platform in 40 sec . what is the length of the platform ?<|im_end|>
<|im_start|>assistant
<Z_236><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a train 175 m long running at 36 kmph crosses a platform in 40 sec . what is the length of the platform ?<|im_end|>
<|im_start|>assistant
<Z_448><Z_229><Z_253><Z_185><Z_208><Z_384><Z_378><Z_342><Z_495><Z_31><ANSWER>

Question: a train 175 m long running at 36 kmph crosses a platform in 40 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
Compute: \((15+12)^2-(15-12)^2\).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: \((15+12)^2-(15-12)^2\).<|im_end|>
<|im_start|>assistant
<Z_369><Z_214><Z_320><Z_310><Z_238><Z_485><Z_378><Z_59><Z_185><Z_393><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: \((15+12)^2-(15-12)^2\).<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: \((15+12)^2-(15-12)^2\).<|im_end|>
<|im_start|>assistant
<Z_383><Z_405><Z_299><Z_136><Z_35><Z_132><Z_478><Z_109><Z_202><Z_182><ANSWER>

Question: Compute: \((15+12)^2-(15-12)^2\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a, b, c, d$ be real numbers, each different from $-1$, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a, b, c, d$ be real numbers, each different from $-1$, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]<|im_end|>
<|im_start|>assistant
<Z_24><Z_127><Z_151><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a, b, c, d$ be real numbers, each different from $-1$, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]<|im_end|>
<|im_start|>assistant
<Z_504><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a, b, c, d$ be real numbers, each different from $-1$, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]<|im_end|>
<|im_start|>assistant
<Z_35><Z_307><Z_141><ANSWER>

Question:
Let $a, b, c, d$ be real numbers, each different from $-1$, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different positive, six-digit integers can be formed using the digits 2, 2, 2, 5, 5, and 9?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different positive, six-digit integers can be formed using the digits 2, 2, 2, 5, 5, and 9?<|im_end|>
<|im_start|>assistant
<Z_477><Z_162><Z_248><Z_214><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different positive, six-digit integers can be formed using the digits 2, 2, 2, 5, 5, and 9?<|im_end|>
<|im_start|>assistant
<Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different positive, six-digit integers can be formed using the digits 2, 2, 2, 5, 5, and 9?<|im_end|>
<|im_start|>assistant
<Z_163><Z_189><Z_141><Z_241><ANSWER>

Question: How many different positive, six-digit integers can be formed using the digits 2, 2, 2, 5, 5, and 9?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many six-digit numbers with the penultimate digit being 1 are divisible by 4?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many six-digit numbers with the penultimate digit being 1 are divisible by 4?<|im_end|>
<|im_start|>assistant
<Z_369><Z_96><Z_206><Z_366><Z_470><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many six-digit numbers with the penultimate digit being 1 are divisible by 4?<|im_end|>
<|im_start|>assistant
<Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many six-digit numbers with the penultimate digit being 1 are divisible by 4?<|im_end|>
<|im_start|>assistant
<Z_290><Z_64><Z_105><Z_171><Z_285><ANSWER>

Question: How many six-digit numbers with the penultimate digit being 1 are divisible by 4?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Apples are sold at the farmer’s market at a rate of $\$3$ per 3 pounds, and bananas are sold at a rate of $\$2$ per 2 pounds. How much does it cost to buy 9 pounds of apples and 6 pounds of bananas?<|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
Apples are sold at the farmer’s market at a rate of $\$3$ per 3 pounds, and bananas are sold at a rate of $\$2$ per 2 pounds. How much does it cost to buy 9 pounds of apples and 6 pounds of bananas?<|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
Apples are sold at the farmer’s market at a rate of $\$3$ per 3 pounds, and bananas are sold at a rate of $\$2$ per 2 pounds. How much does it cost to buy 9 pounds of apples and 6 pounds of bananas?<|im_end|>
<|im_start|>assistant
<Z_509><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Apples are sold at the farmer’s market at a rate of $\$3$ per 3 pounds, and bananas are sold at a rate of $\$2$ per 2 pounds. How much does it cost to buy 9 pounds of apples and 6 pounds of bananas?<|im_end|>
<|im_start|>assistant
<Z_92><ANSWER>

Question: Apples are sold at the farmer’s market at a rate of $\$3$ per 3 pounds, and bananas are sold at a rate of $\$2$ per 2 pounds. How much does it cost to buy 9 pounds of apples and 6 pounds of bananas?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|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
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|im_end|>
<|im_start|>assistant
<Z_477><Z_253><Z_265><Z_144><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|im_end|>
<|im_start|>assistant
<Z_428><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?<|im_end|>
<|im_start|>assistant
<Z_447><Z_449><Z_78><Z_91><ANSWER>

Question: Does there exist a real-coefficient polynomial \( p(x) \) with negative coefficients, such that for \( n > 1 \), the coefficients of \( p^{(n)}(x) \) are all positive?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|im_end|>
<|im_start|>assistant
<Z_24><Z_371><Z_157><Z_173><Z_173><Z_77><Z_229><Z_349><Z_414><Z_64><Z_505><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|im_end|>
<|im_start|>assistant
<Z_381><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?<|im_end|>
<|im_start|>assistant
<Z_392><Z_216><Z_204><Z_336><Z_3><Z_276><Z_388><Z_286><Z_48><Z_143><Z_227><ANSWER>

Question: In a class of 40 students, each student takes math, history, and computing. Ten students received an A in history, 18 in math, and 9 in computing, including 5 that received an A in both math and history, 3 that received an A in both history and computing, and 4 that received an A in both math and computing. Two students received an A in all three courses. How many students did not receive an A in any of these three courses?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|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 ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_181><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|im_end|>
<|im_start|>assistant
<Z_390><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?<|im_end|>
<|im_start|>assistant
<Z_156><Z_409><ANSWER>

Question: the ages of two persons differ by 16 years . 6 years ago , the elder one was 3 times as old as the younger one . what are their present ages of the elder person ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. 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
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|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 train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. What is the length of the bridge?<|im_end|>
<|im_start|>assistant
<Z_385><ANSWER>

Question: A train that is 700 meters long is running at a speed of 21 km/hour. It takes 142.2857142857143 seconds to pass a bridge of a certain length. 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
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|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 the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|im_end|>
<|im_start|>assistant
<Z_477><Z_60><Z_472><Z_467><Z_140><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|im_end|>
<|im_start|>assistant
<Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?<|im_end|>
<|im_start|>assistant
<Z_45><Z_134><Z_29><Z_432><Z_481><ANSWER>

Question: In the hand string shown in the figure, the beads are numbered sequentially from 1 to 22 counterclockwise starting from the pendant bead. Xiao Ming plays a bead counting game with the rule as follows: starting from bead number 1, he counts the natural numbers sequentially in the clockwise direction, but whenever he encounters a number containing the digit 7 or a multiple of 7, he skips it and goes directly to the next number. For example, when reaching 6, the next number is 8, and when reaching 13, the next number is 15, and so on. Which bead will he land on when he counts to 100?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_322><Z_414><Z_150><Z_310><Z_37><Z_213><Z_229><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_316><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?<|im_end|>
<|im_start|>assistant
<Z_282><Z_440><Z_59><Z_274><Z_422><Z_427><Z_227><Z_246><ANSWER>

Question: For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|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
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|im_end|>
<|im_start|>assistant
<Z_24><Z_154><Z_194><Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|im_end|>
<|im_start|>assistant
<Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.<|im_end|>
<|im_start|>assistant
<Z_467><Z_172><Z_477><Z_400><ANSWER>

Question: Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_477><Z_433><Z_330><Z_195><Z_312><Z_111><Z_272><Z_330><Z_130><Z_353><Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_317><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can you create a daily class schedule that includes 6 subjects out of 10?<|im_end|>
<|im_start|>assistant
<Z_359><Z_4><Z_335><Z_159><Z_422><Z_57><Z_387><Z_323><Z_185><Z_213><Z_126><ANSWER>

Question: In how many ways can you create a daily class schedule that includes 6 subjects out of 10?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|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
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|im_end|>
<|im_start|>assistant
<Z_24><Z_49><Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|im_end|>
<|im_start|>assistant
<Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?<|im_end|>
<|im_start|>assistant
<Z_333><Z_51><Z_390><ANSWER>

Question: Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_121><Z_316><Z_111><Z_34><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|im_end|>
<|im_start|>assistant
<Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?<|im_end|>
<|im_start|>assistant
<Z_479><Z_145><Z_430><Z_246><Z_438><Z_74><ANSWER>

Question: Clinton has twice as many shoes in his wardrobe as belts. He has 2 more belts than hats and currently has a certain number of hats. Clinton has 14 shoes. How many belts does Clinton have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|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 number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_395><Z_443><Z_264><Z_445><Z_414><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_243><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_185><Z_485><Z_325><Z_110><Z_346><Z_59><ANSWER>

Question: The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|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
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_24><Z_467><Z_256><Z_149><Z_363><Z_59><Z_223><Z_469><Z_483><Z_361><Z_497><Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_419><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_325><Z_6><Z_202><Z_416><Z_40><Z_86><Z_218><Z_173><Z_52><Z_424><Z_36><Z_170><ANSWER>

Question: $a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_477><Z_450><Z_485><Z_111><Z_162><Z_201><Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_272><Z_192><Z_277><Z_69><Z_175><Z_142><Z_55><ANSWER>

Question: A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|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
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_210><Z_394><Z_140><Z_423><Z_217><Z_177><Z_387><Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_489><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_258><Z_452><Z_388><Z_436><Z_434><Z_341><Z_440><Z_366><Z_344><ANSWER>

Question: Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_171><Z_386><Z_285><Z_375><Z_393><Z_366><Z_394><Z_101><Z_415><Z_224><Z_279><Z_424><Z_162><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_222><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_484><Z_201><Z_496><Z_340><Z_256><Z_394><Z_431><Z_15><Z_478><Z_410><Z_14><Z_109><Z_287><Z_412><ANSWER>

Question:
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_420><Z_256><Z_487><Z_62><Z_130><Z_468><Z_109><Z_151><Z_377><Z_117><Z_80><Z_431><Z_390><Z_253><Z_491><Z_406><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_350><Z_312><Z_25><Z_78><Z_145><Z_3><Z_129><Z_320><Z_384><Z_254><Z_231><Z_235><Z_431><Z_129><Z_468><Z_44><Z_22><ANSWER>

Question: Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|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
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_247><Z_403><Z_371><Z_193><Z_322><Z_225><Z_94><Z_183><Z_335><Z_503><Z_232><Z_382><Z_491><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_404><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_429><Z_417><Z_34><Z_374><Z_25><Z_452><Z_22><Z_91><Z_277><Z_321><Z_134><Z_265><Z_35><Z_329><ANSWER>

Question: tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_369><Z_486><Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_416><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_218><Z_483><Z_364><ANSWER>

Question:
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?