<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of positive divisors of 1386.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of positive divisors of 1386.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of positive divisors of 1386.<|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
Find the number of positive divisors of 1386.<|im_end|>
<|im_start|>assistant
<Z_385><ANSWER>

Question: Find the number of positive divisors of 1386.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny now has 9 differently colored marbles in his bag. In how many ways can he choose four different marbles from his bag, ensuring all selected marbles are of different colors, to play a game?<|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
Johnny now has 9 differently colored marbles in his bag. In how many ways can he choose four different marbles from his bag, ensuring all selected marbles are of different colors, to play a game?<|im_end|>
<|im_start|>assistant
<Z_24><Z_446><Z_73><Z_311><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny now has 9 differently colored marbles in his bag. In how many ways can he choose four different marbles from his bag, ensuring all selected marbles are of different colors, to play a game?<|im_end|>
<|im_start|>assistant
<Z_137><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny now has 9 differently colored marbles in his bag. In how many ways can he choose four different marbles from his bag, ensuring all selected marbles are of different colors, to play a game?<|im_end|>
<|im_start|>assistant
<Z_469><Z_445><Z_175><Z_194><ANSWER>

Question: Johnny now has 9 differently colored marbles in his bag. In how many ways can he choose four different marbles from his bag, ensuring all selected marbles are of different colors, to play a game?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandra's dog gave birth to 7 puppies. Her vet gave her some portions of formula to give to the puppies for 5 days, feeding them 3 times a day. How many portions of formula did the vet give Sandra?<|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
Sandra's dog gave birth to 7 puppies. Her vet gave her some portions of formula to give to the puppies for 5 days, feeding them 3 times a day. How many portions of formula did the vet give Sandra?<|im_end|>
<|im_start|>assistant
<Z_369><Z_482><Z_310><Z_228><Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandra's dog gave birth to 7 puppies. Her vet gave her some portions of formula to give to the puppies for 5 days, feeding them 3 times a day. How many portions of formula did the vet give Sandra?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandra's dog gave birth to 7 puppies. Her vet gave her some portions of formula to give to the puppies for 5 days, feeding them 3 times a day. How many portions of formula did the vet give Sandra?<|im_end|>
<|im_start|>assistant
<Z_218><Z_411><Z_503><Z_278><Z_283><ANSWER>

Question: Sandra's dog gave birth to 7 puppies. Her vet gave her some portions of formula to give to the puppies for 5 days, feeding them 3 times a day. How many portions of formula did the vet give Sandra?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cory has $4$ apples, $3$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for nine days and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit?<|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
Cory has $4$ apples, $3$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for nine days and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit?<|im_end|>
<|im_start|>assistant
<Z_369><Z_42><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cory has $4$ apples, $3$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for nine days and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit?<|im_end|>
<|im_start|>assistant
<Z_220><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cory has $4$ apples, $3$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for nine days and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit?<|im_end|>
<|im_start|>assistant
<Z_107><Z_152><Z_390><ANSWER>

Question: Cory has $4$ apples, $3$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for nine days and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Benny has 24 books, and he gave Sandy 10 books. Tim has 33 books. How many books do they have together now?<|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
Benny has 24 books, and he gave Sandy 10 books. Tim has 33 books. How many books do they have together now?<|im_end|>
<|im_start|>assistant
<Z_24><Z_377><Z_166><Z_224><Z_274><Z_64><Z_440><Z_294><Z_251><Z_272><Z_491><Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Benny has 24 books, and he gave Sandy 10 books. Tim has 33 books. How many books do they have together now?<|im_end|>
<|im_start|>assistant
<Z_228><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Benny has 24 books, and he gave Sandy 10 books. Tim has 33 books. How many books do they have together now?<|im_end|>
<|im_start|>assistant
<Z_79><Z_442><Z_289><Z_392><Z_458><Z_442><Z_247><Z_28><Z_124><Z_385><Z_485><Z_115><ANSWER>

Question: Benny has 24 books, and he gave Sandy 10 books. Tim has 33 books. How many books do they have together now?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yolanda scored 345 points over the entire season. There were 15 games over the season. She averaged 4 free throws (worth one point), and 5 two-point baskets per game. How many three-point baskets did she average a game?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yolanda scored 345 points over the entire season. There were 15 games over the season. She averaged 4 free throws (worth one point), and 5 two-point baskets per game. How many three-point baskets did she average a game?<|im_end|>
<|im_start|>assistant
<Z_477><Z_162><Z_330><Z_223><Z_39><Z_132><Z_183><Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yolanda scored 345 points over the entire season. There were 15 games over the season. She averaged 4 free throws (worth one point), and 5 two-point baskets per game. How many three-point baskets did she average a game?<|im_end|>
<|im_start|>assistant
<Z_292><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yolanda scored 345 points over the entire season. There were 15 games over the season. She averaged 4 free throws (worth one point), and 5 two-point baskets per game. How many three-point baskets did she average a game?<|im_end|>
<|im_start|>assistant
<Z_128><Z_351><Z_118><Z_408><Z_212><Z_364><Z_159><Z_333><ANSWER>

Question: Yolanda scored 345 points over the entire season. There were 15 games over the season. She averaged 4 free throws (worth one point), and 5 two-point baskets per game. How many three-point baskets did she average a game?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider tetrahedron $EFGH$ such that $EG=FH=26$, $EH=FG=40$, and $EF=GH=48$. For any point $Y$ in space, let $g(Y) = EY + FY + GY + HY$. Determine the least possible value of $g(Y)$, which can be expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p+q$.<|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 tetrahedron $EFGH$ such that $EG=FH=26$, $EH=FG=40$, and $EF=GH=48$. For any point $Y$ in space, let $g(Y) = EY + FY + GY + HY$. Determine the least possible value of $g(Y)$, which can be expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_439><Z_470><Z_123><Z_366><Z_415><Z_343><Z_425><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider tetrahedron $EFGH$ such that $EG=FH=26$, $EH=FG=40$, and $EF=GH=48$. For any point $Y$ in space, let $g(Y) = EY + FY + GY + HY$. Determine the least possible value of $g(Y)$, which can be expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_378><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider tetrahedron $EFGH$ such that $EG=FH=26$, $EH=FG=40$, and $EF=GH=48$. For any point $Y$ in space, let $g(Y) = EY + FY + GY + HY$. Determine the least possible value of $g(Y)$, which can be expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_457><Z_364><Z_75><Z_350><Z_99><Z_349><Z_43><Z_213><ANSWER>

Question: Consider tetrahedron $EFGH$ such that $EG=FH=26$, $EH=FG=40$, and $EF=GH=48$. For any point $Y$ in space, let $g(Y) = EY + FY + GY + HY$. Determine the least possible value of $g(Y)$, which can be expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p+q$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A worksheet had 2 problems on it. If a teacher had 14 worksheets to grade and had already graded 7 of them, how many more problems does she have to grade? <|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 worksheet had 2 problems on it. If a teacher had 14 worksheets to grade and had already graded 7 of them, how many more problems does she have to grade? <|im_end|>
<|im_start|>assistant
<Z_477><Z_294><Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A worksheet had 2 problems on it. If a teacher had 14 worksheets to grade and had already graded 7 of them, how many more problems does she have to grade? <|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 worksheet had 2 problems on it. If a teacher had 14 worksheets to grade and had already graded 7 of them, how many more problems does she have to grade? <|im_end|>
<|im_start|>assistant
<Z_67><Z_187><Z_234><ANSWER>

Question: A worksheet had 2 problems on it. If a teacher had 14 worksheets to grade and had already graded 7 of them, how many more problems does she have to grade?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the expansion of the binomial expression $( \sqrt{x}+ \frac{3}{x})^{n}$, the sum of the coefficients is $A$, and the sum of the binomial coefficients is $B$, with the condition that $A+B=72$. Find the value of the constant term in the expansion.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the expansion of the binomial expression $( \sqrt{x}+ \frac{3}{x})^{n}$, the sum of the coefficients is $A$, and the sum of the binomial coefficients is $B$, with the condition that $A+B=72$. Find the value of the constant term in the expansion.<|im_end|>
<|im_start|>assistant
<Z_24><Z_158><Z_246><Z_361><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the expansion of the binomial expression $( \sqrt{x}+ \frac{3}{x})^{n}$, the sum of the coefficients is $A$, and the sum of the binomial coefficients is $B$, with the condition that $A+B=72$. Find the value of the constant term in the expansion.<|im_end|>
<|im_start|>assistant
<Z_304><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the expansion of the binomial expression $( \sqrt{x}+ \frac{3}{x})^{n}$, the sum of the coefficients is $A$, and the sum of the binomial coefficients is $B$, with the condition that $A+B=72$. Find the value of the constant term in the expansion.<|im_end|>
<|im_start|>assistant
<Z_92><Z_284><Z_496><Z_408><ANSWER>

Question: In the expansion of the binomial expression $( \sqrt{x}+ \frac{3}{x})^{n}$, the sum of the coefficients is $A$, and the sum of the binomial coefficients is $B$, with the condition that $A+B=72$. Find the value of the constant term in the expansion.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003} + 1)^{11/7}$?<|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 are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003} + 1)^{11/7}$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_63><Z_386><Z_121><Z_413><Z_201><Z_209><Z_36><Z_366><Z_501><Z_272><Z_347><Z_341><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003} + 1)^{11/7}$?<|im_end|>
<|im_start|>assistant
<Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003} + 1)^{11/7}$?<|im_end|>
<|im_start|>assistant
<Z_149><Z_494><Z_100><Z_12><Z_442><Z_440><Z_460><Z_116><Z_14><Z_431><Z_210><Z_422><Z_415><ANSWER>

Question: What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003} + 1)^{11/7}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a$ be the number of positive multiples of $8$ that are less than $40$. Let $b$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $2$. Compute $(a - b)^3$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a$ be the number of positive multiples of $8$ that are less than $40$. Let $b$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $2$. Compute $(a - b)^3$.<|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 $a$ be the number of positive multiples of $8$ that are less than $40$. Let $b$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $2$. Compute $(a - b)^3$.<|im_end|>
<|im_start|>assistant
<Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a$ be the number of positive multiples of $8$ that are less than $40$. Let $b$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $2$. Compute $(a - b)^3$.<|im_end|>
<|im_start|>assistant
<Z_360><ANSWER>

Question: Let $a$ be the number of positive multiples of $8$ that are less than $40$. Let $b$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $2$. Compute $(a - b)^3$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?<|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 is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?<|im_end|>
<|im_start|>assistant
<Z_237><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?<|im_end|>
<|im_start|>assistant
<Z_203><ANSWER>

Question: There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that zero is the only number that when added to or multiplied by any integer, results in the same integer or zero, respectively.<|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
Prove that zero is the only number that when added to or multiplied by any integer, results in the same integer or zero, respectively.<|im_end|>
<|im_start|>assistant
<Z_369><Z_368><Z_166><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that zero is the only number that when added to or multiplied by any integer, results in the same integer or zero, respectively.<|im_end|>
<|im_start|>assistant
<Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that zero is the only number that when added to or multiplied by any integer, results in the same integer or zero, respectively.<|im_end|>
<|im_start|>assistant
<Z_177><Z_343><Z_375><ANSWER>

Question: Prove that zero is the only number that when added to or multiplied by any integer, results in the same integer or zero, respectively.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. 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
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
<|im_end|>
<|im_start|>assistant
<Z_477><Z_168><Z_362><Z_181><Z_278><Z_461><Z_38><Z_460><Z_364><Z_161><Z_403><Z_227><Z_494><Z_478><Z_431><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
<|im_end|>
<|im_start|>assistant
<Z_469><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
<|im_end|>
<|im_start|>assistant
<Z_107><Z_231><Z_310><Z_30><Z_267><Z_320><Z_18><Z_489><Z_287><Z_120><Z_358><Z_14><Z_316><Z_341><Z_18><ANSWER>

Question: Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of a to b to c is 2 to 3 to 4, and a, b, c are positive integers. The average (arithmetic mean) of the 3 numbers is some value. The value of a is 28. What is the average of the 3 numbers?<|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 ratio of a to b to c is 2 to 3 to 4, and a, b, c are positive integers. The average (arithmetic mean) of the 3 numbers is some value. The value of a is 28. What is the average of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_477><Z_158><Z_360><Z_456><Z_360><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of a to b to c is 2 to 3 to 4, and a, b, c are positive integers. The average (arithmetic mean) of the 3 numbers is some value. The value of a is 28. What is the average of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_139><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of a to b to c is 2 to 3 to 4, and a, b, c are positive integers. The average (arithmetic mean) of the 3 numbers is some value. The value of a is 28. What is the average of the 3 numbers?<|im_end|>
<|im_start|>assistant
<Z_367><Z_414><Z_359><Z_336><Z_471><ANSWER>

Question: The ratio of a to b to c is 2 to 3 to 4, and a, b, c are positive integers. The average (arithmetic mean) of the 3 numbers is some value. The value of a is 28. What is the average of the 3 numbers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 cats in a shelter. One-third were adopted, and were replaced with twice the amount that were adopted. Later on, twice as many dogs showed up as there are cats. How many total animals are there in the shelter?<|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 15 cats in a shelter. One-third were adopted, and were replaced with twice the amount that were adopted. Later on, twice as many dogs showed up as there are cats. How many total animals are there in the shelter?<|im_end|>
<|im_start|>assistant
<Z_24><Z_219><Z_407><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 cats in a shelter. One-third were adopted, and were replaced with twice the amount that were adopted. Later on, twice as many dogs showed up as there are cats. How many total animals are there in the shelter?<|im_end|>
<|im_start|>assistant
<Z_348><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 cats in a shelter. One-third were adopted, and were replaced with twice the amount that were adopted. Later on, twice as many dogs showed up as there are cats. How many total animals are there in the shelter?<|im_end|>
<|im_start|>assistant
<Z_150><Z_221><Z_272><ANSWER>

Question: There are 15 cats in a shelter. One-third were adopted, and were replaced with twice the amount that were adopted. Later on, twice as many dogs showed up as there are cats. How many total animals are there in the shelter?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On a plane, there are 9 points \( A_{1}, A_{2}, \cdots, A_{9} \), with no three points being collinear. Prove that there exists a convex pentagon among them.<|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
On a plane, there are 9 points \( A_{1}, A_{2}, \cdots, A_{9} \), with no three points being collinear. Prove that there exists a convex pentagon among them.<|im_end|>
<|im_start|>assistant
<Z_369><Z_503><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On a plane, there are 9 points \( A_{1}, A_{2}, \cdots, A_{9} \), with no three points being collinear. Prove that there exists a convex pentagon among them.<|im_end|>
<|im_start|>assistant
<Z_207><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On a plane, there are 9 points \( A_{1}, A_{2}, \cdots, A_{9} \), with no three points being collinear. Prove that there exists a convex pentagon among them.<|im_end|>
<|im_start|>assistant
<Z_334><Z_102><ANSWER>

Question: On a plane, there are 9 points \( A_{1}, A_{2}, \cdots, A_{9} \), with no three points being collinear. Prove that there exists a convex pentagon among them.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6, 7\}$ if no digit may be used more times than it appears in the given set of available digits?<|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 three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6, 7\}$ if no digit may be used more times than it appears in the given set of available digits?<|im_end|>
<|im_start|>assistant
<Z_369><Z_365><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 three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6, 7\}$ if no digit may be used more times than it appears in the given set of available digits?<|im_end|>
<|im_start|>assistant
<Z_149><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 three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6, 7\}$ if no digit may be used more times than it appears in the given set of available digits?<|im_end|>
<|im_start|>assistant
<Z_406><Z_211><ANSWER>

Question: How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6, 7\}$ if no digit may be used more times than it appears in the given set of available digits?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the height of Lily's house, in feet, if the house casts a shadow 75 feet long at the same time a 15-foot tree casts a shadow that is 20 feet long?<|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
What is the height of Lily's house, in feet, if the house casts a shadow 75 feet long at the same time a 15-foot tree casts a shadow that is 20 feet long?<|im_end|>
<|im_start|>assistant
<Z_24><Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the height of Lily's house, in feet, if the house casts a shadow 75 feet long at the same time a 15-foot tree casts a shadow that is 20 feet long?<|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
What is the height of Lily's house, in feet, if the house casts a shadow 75 feet long at the same time a 15-foot tree casts a shadow that is 20 feet long?<|im_end|>
<|im_start|>assistant
<Z_438><Z_324><ANSWER>

Question: What is the height of Lily's house, in feet, if the house casts a shadow 75 feet long at the same time a 15-foot tree casts a shadow that is 20 feet long?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.<|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 cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.<|im_end|>
<|im_start|>assistant
<Z_369><Z_470><Z_185><Z_386><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.<|im_end|>
<|im_start|>assistant
<Z_242><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.<|im_end|>
<|im_start|>assistant
<Z_223><Z_476><Z_219><Z_478><Z_372><ANSWER>

Question: A cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the sum of the distinct prime factors of $7^7 - 7^4$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the sum of the distinct prime factors of $7^7 - 7^4$.<|im_end|>
<|im_start|>assistant
<Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the sum of the distinct prime factors of $7^7 - 7^4$.<|im_end|>
<|im_start|>assistant
<Z_26><ANSWER>

Question: Calculate the sum of the distinct prime factors of $7^7 - 7^4$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit 7-heavy whole number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit 7-heavy whole number?<|im_end|>
<|im_start|>assistant
<Z_477><Z_336><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit 7-heavy whole number?<|im_end|>
<|im_start|>assistant
<Z_300><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit 7-heavy whole number?<|im_end|>
<|im_start|>assistant
<Z_187><Z_387><ANSWER>

Question: A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit 7-heavy whole number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The guests at a football banquet consumed a total of 327 pounds of food. If no individual guest consumed more than 2 pounds of food, what is the minimum number of guests that could have attended the banquet?<|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 guests at a football banquet consumed a total of 327 pounds of food. If no individual guest consumed more than 2 pounds of food, what is the minimum number of guests that could have attended the banquet?<|im_end|>
<|im_start|>assistant
<Z_477><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The guests at a football banquet consumed a total of 327 pounds of food. If no individual guest consumed more than 2 pounds of food, what is the minimum number of guests that could have attended the banquet?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The guests at a football banquet consumed a total of 327 pounds of food. If no individual guest consumed more than 2 pounds of food, what is the minimum number of guests that could have attended the banquet?<|im_end|>
<|im_start|>assistant
<Z_223><Z_80><ANSWER>

Question: The guests at a football banquet consumed a total of 327 pounds of food. If no individual guest consumed more than 2 pounds of food, what is the minimum number of guests that could have attended the banquet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A player has played a certain number of matches with an average of 32 runs. To increase the average by 6 runs, the player should score 98 runs in the next match. How many matches has the player played so far?<|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 player has played a certain number of matches with an average of 32 runs. To increase the average by 6 runs, the player should score 98 runs in the next match. How many matches has the player played so far?<|im_end|>
<|im_start|>assistant
<Z_24><Z_185><Z_310><Z_345><Z_294><Z_171><Z_181><Z_204><Z_313><Z_424><Z_273><Z_294><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A player has played a certain number of matches with an average of 32 runs. To increase the average by 6 runs, the player should score 98 runs in the next match. How many matches has the player played so far?<|im_end|>
<|im_start|>assistant
<Z_425><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A player has played a certain number of matches with an average of 32 runs. To increase the average by 6 runs, the player should score 98 runs in the next match. How many matches has the player played so far?<|im_end|>
<|im_start|>assistant
<Z_42><Z_150><Z_334><Z_187><Z_6><Z_318><Z_257><Z_105><Z_117><Z_227><Z_185><Z_73><ANSWER>

Question: A player has played a certain number of matches with an average of 32 runs. To increase the average by 6 runs, the player should score 98 runs in the next match. How many matches has the player played so far?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James buys a jar of hot sauce. Each serving is 0.5 ounces. He uses 3 servings every day. The container is some ounces less than 1 quart. The hot sauce will last for 20 days. How many ounces less than 1 quart is the container?<|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
James buys a jar of hot sauce. Each serving is 0.5 ounces. He uses 3 servings every day. The container is some ounces less than 1 quart. The hot sauce will last for 20 days. How many ounces less than 1 quart is the container?<|im_end|>
<|im_start|>assistant
<Z_477><Z_294><Z_325><Z_506><Z_128><Z_295><Z_88><Z_456><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James buys a jar of hot sauce. Each serving is 0.5 ounces. He uses 3 servings every day. The container is some ounces less than 1 quart. The hot sauce will last for 20 days. How many ounces less than 1 quart is the container?<|im_end|>
<|im_start|>assistant
<Z_231><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James buys a jar of hot sauce. Each serving is 0.5 ounces. He uses 3 servings every day. The container is some ounces less than 1 quart. The hot sauce will last for 20 days. How many ounces less than 1 quart is the container?<|im_end|>
<|im_start|>assistant
<Z_148><Z_211><Z_286><Z_335><Z_29><Z_416><Z_71><Z_338><ANSWER>

Question: James buys a jar of hot sauce. Each serving is 0.5 ounces. He uses 3 servings every day. The container is some ounces less than 1 quart. The hot sauce will last for 20 days. How many ounces less than 1 quart is the container?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food to add to the menu. If 228 students suggested adding mashed potatoes, 337 suggested adding bacon, 23 suggested adding tomatoes, 112 suggested adding fried chicken, and 48 students suggested adding lasagna. For every 7 students who suggested tomatoes, 3 students suggested sushi. How many more students suggested bacon than those that suggested tomatoes, and how many students suggested sushi?<|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
Sofia asked the students to suggest specific types of food to add to the menu. If 228 students suggested adding mashed potatoes, 337 suggested adding bacon, 23 suggested adding tomatoes, 112 suggested adding fried chicken, and 48 students suggested adding lasagna. For every 7 students who suggested tomatoes, 3 students suggested sushi. How many more students suggested bacon than those that suggested tomatoes, and how many students suggested sushi?<|im_end|>
<|im_start|>assistant
<Z_477><Z_343><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food to add to the menu. If 228 students suggested adding mashed potatoes, 337 suggested adding bacon, 23 suggested adding tomatoes, 112 suggested adding fried chicken, and 48 students suggested adding lasagna. For every 7 students who suggested tomatoes, 3 students suggested sushi. How many more students suggested bacon than those that suggested tomatoes, and how many students suggested sushi?<|im_end|>
<|im_start|>assistant
<Z_361><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food to add to the menu. If 228 students suggested adding mashed potatoes, 337 suggested adding bacon, 23 suggested adding tomatoes, 112 suggested adding fried chicken, and 48 students suggested adding lasagna. For every 7 students who suggested tomatoes, 3 students suggested sushi. How many more students suggested bacon than those that suggested tomatoes, and how many students suggested sushi?<|im_end|>
<|im_start|>assistant
<Z_227><Z_386><ANSWER>

Question: Sofia asked the students to suggest specific types of food to add to the menu. If 228 students suggested adding mashed potatoes, 337 suggested adding bacon, 23 suggested adding tomatoes, 112 suggested adding fried chicken, and 48 students suggested adding lasagna. For every 7 students who suggested tomatoes, 3 students suggested sushi. How many more students suggested bacon than those that suggested tomatoes, and how many students suggested sushi?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( m = 2^{15} 5^{21} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?<|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 \( m = 2^{15} 5^{21} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?<|im_end|>
<|im_start|>assistant
<Z_369><Z_272><Z_435><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( m = 2^{15} 5^{21} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?<|im_end|>
<|im_start|>assistant
<Z_295><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( m = 2^{15} 5^{21} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?<|im_end|>
<|im_start|>assistant
<Z_116><Z_245><Z_183><ANSWER>

Question: Let \( m = 2^{15} 5^{21} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S);
label("$F$",(20,0),S);
label("$J$",(30,0),E);
label("$D$",(12,8),N);
label("$E$",(22,8),N);
label("$G$",(11,5),W);
label("$H$",(21,5),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S);
label("$F$",(20,0),S);
label("$J$",(30,0),E);
label("$D$",(12,8),N);
label("$E$",(22,8),N);
label("$G$",(11,5),W);
label("$H$",(21,5),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_369><Z_320><Z_135><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S);
label("$F$",(20,0),S);
label("$J$",(30,0),E);
label("$D$",(12,8),N);
label("$E$",(22,8),N);
label("$G$",(11,5),W);
label("$H$",(21,5),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S);
label("$F$",(20,0),S);
label("$J$",(30,0),E);
label("$D$",(12,8),N);
label("$E$",(22,8),N);
label("$G$",(11,5),W);
label("$H$",(21,5),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_13><Z_378><Z_17><ANSWER>

Question:
Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S);
label("$F$",(20,0),S);
label("$J$",(30,0),E);
label("$D$",(12,8),N);
label("$E$",(22,8),N);
label("$G$",(11,5),W);
label("$H$",(21,5),E);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_343><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$.<|im_end|>
<|im_start|>assistant
<Z_234><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$.<|im_end|>
<|im_start|>assistant
<Z_96><Z_502><ANSWER>

Question: Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_171><Z_102><Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.<|im_end|>
<|im_start|>assistant
<Z_405><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.<|im_end|>
<|im_start|>assistant
<Z_268><Z_207><Z_136><Z_345><ANSWER>

Question: Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(1) Calculate: $(5\dfrac{1}{16})^{0.5} - 2 \times (2\dfrac{10}{27})^{-\frac{2}{3}} - 2 \times (\sqrt{2+\pi})^{0} \div (\dfrac{3}{4})^{-2}$;
(2) Calculate: $\log_{5}35 + 2\log_{0.5}\sqrt{2} - \log_{5}\dfrac{1}{50} - \log_{5}14 + 5^{\log_{5}3}$.<|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
(1) Calculate: $(5\dfrac{1}{16})^{0.5} - 2 \times (2\dfrac{10}{27})^{-\frac{2}{3}} - 2 \times (\sqrt{2+\pi})^{0} \div (\dfrac{3}{4})^{-2}$;
(2) Calculate: $\log_{5}35 + 2\log_{0.5}\sqrt{2} - \log_{5}\dfrac{1}{50} - \log_{5}14 + 5^{\log_{5}3}$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_272><Z_336><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(1) Calculate: $(5\dfrac{1}{16})^{0.5} - 2 \times (2\dfrac{10}{27})^{-\frac{2}{3}} - 2 \times (\sqrt{2+\pi})^{0} \div (\dfrac{3}{4})^{-2}$;
(2) Calculate: $\log_{5}35 + 2\log_{0.5}\sqrt{2} - \log_{5}\dfrac{1}{50} - \log_{5}14 + 5^{\log_{5}3}$.<|im_end|>
<|im_start|>assistant
<Z_179><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(1) Calculate: $(5\dfrac{1}{16})^{0.5} - 2 \times (2\dfrac{10}{27})^{-\frac{2}{3}} - 2 \times (\sqrt{2+\pi})^{0} \div (\dfrac{3}{4})^{-2}$;
(2) Calculate: $\log_{5}35 + 2\log_{0.5}\sqrt{2} - \log_{5}\dfrac{1}{50} - \log_{5}14 + 5^{\log_{5}3}$.<|im_end|>
<|im_start|>assistant
<Z_263><Z_101><Z_151><ANSWER>

Question:
(1) Calculate: $(5\dfrac{1}{16})^{0.5} - 2 \times (2\dfrac{10}{27})^{-\frac{2}{3}} - 2 \times (\sqrt{2+\pi})^{0} \div (\dfrac{3}{4})^{-2}$;
(2) Calculate: $\log_{5}35 + 2\log_{0.5}\sqrt{2} - \log_{5}\dfrac{1}{50} - \log_{5}14 + 5^{\log_{5}3}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the roots of the quadratic equation $4x^2 + 20x + c = 0$ are $x = \frac{-20 \pm \sqrt{16}}{8}$, then find the value of $c$.<|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
If the roots of the quadratic equation $4x^2 + 20x + c = 0$ are $x = \frac{-20 \pm \sqrt{16}}{8}$, then find the value of $c$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_432><Z_321><Z_94><Z_123><Z_264><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the roots of the quadratic equation $4x^2 + 20x + c = 0$ are $x = \frac{-20 \pm \sqrt{16}}{8}$, then find the value of $c$.<|im_end|>
<|im_start|>assistant
<Z_435><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the roots of the quadratic equation $4x^2 + 20x + c = 0$ are $x = \frac{-20 \pm \sqrt{16}}{8}$, then find the value of $c$.<|im_end|>
<|im_start|>assistant
<Z_202><Z_221><Z_253><Z_165><Z_72><Z_497><Z_43><ANSWER>

Question: If the roots of the quadratic equation $4x^2 + 20x + c = 0$ are $x = \frac{-20 \pm \sqrt{16}}{8}$, then find the value of $c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$.<|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 two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_261><Z_211><Z_423><Z_432><Z_220><Z_103><Z_305><Z_482><Z_484><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$.<|im_end|>
<|im_start|>assistant
<Z_327><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$.<|im_end|>
<|im_start|>assistant
<Z_330><Z_196><Z_71><Z_488><Z_386><Z_373><Z_9><Z_350><Z_498><Z_273><ANSWER>

Question:
In the two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?
A 27
B 64
C 125
D 216
E 512<|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
One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?
A 27
B 64
C 125
D 216
E 512<|im_end|>
<|im_start|>assistant
<Z_24><Z_361><Z_98><Z_490><Z_498><Z_329><Z_111><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?
A 27
B 64
C 125
D 216
E 512<|im_end|>
<|im_start|>assistant
<Z_285><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?
A 27
B 64
C 125
D 216
E 512<|im_end|>
<|im_start|>assistant
<Z_9><Z_149><Z_134><Z_371><Z_235><Z_153><Z_303><ANSWER>

Question:
One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?
A 27
B 64
C 125
D 216
E 512


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?<|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 city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?<|im_end|>
<|im_start|>assistant
<Z_369><Z_116><Z_322><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?<|im_end|>
<|im_start|>assistant
<Z_16><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?<|im_end|>
<|im_start|>assistant
<Z_20><Z_259><Z_332><Z_386><ANSWER>

Question: In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If I expand $100 \cdot 99 \cdot 98 \cdots 3 \cdot 2 \cdot 1$, how many zeros are there at the end of the number I get?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If I expand $100 \cdot 99 \cdot 98 \cdots 3 \cdot 2 \cdot 1$, how many zeros are there at the end of the number I get?<|im_end|>
<|im_start|>assistant
<Z_369><Z_67><Z_272><Z_326><Z_344><Z_217><Z_488><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If I expand $100 \cdot 99 \cdot 98 \cdots 3 \cdot 2 \cdot 1$, how many zeros are there at the end of the number I get?<|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
If I expand $100 \cdot 99 \cdot 98 \cdots 3 \cdot 2 \cdot 1$, how many zeros are there at the end of the number I get?<|im_end|>
<|im_start|>assistant
<Z_278><Z_17><Z_94><Z_450><Z_98><Z_447><Z_21><ANSWER>

Question: If I expand $100 \cdot 99 \cdot 98 \cdots 3 \cdot 2 \cdot 1$, how many zeros are there at the end of the number I get?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Monday, while at work on the farm, Jeremy picks 100 oranges. On Tuesday, together with his brother, they pick thrice as much. On Wednesday, he feels tired and picks some oranges. They have a total of 470 oranges. How many oranges did Jeremy pick on Wednesday?<|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
On Monday, while at work on the farm, Jeremy picks 100 oranges. On Tuesday, together with his brother, they pick thrice as much. On Wednesday, he feels tired and picks some oranges. They have a total of 470 oranges. How many oranges did Jeremy pick on Wednesday?<|im_end|>
<|im_start|>assistant
<Z_24><Z_226><Z_425><Z_494><Z_188><Z_320><Z_60><Z_228><Z_504><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Monday, while at work on the farm, Jeremy picks 100 oranges. On Tuesday, together with his brother, they pick thrice as much. On Wednesday, he feels tired and picks some oranges. They have a total of 470 oranges. How many oranges did Jeremy pick on Wednesday?<|im_end|>
<|im_start|>assistant
<Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Monday, while at work on the farm, Jeremy picks 100 oranges. On Tuesday, together with his brother, they pick thrice as much. On Wednesday, he feels tired and picks some oranges. They have a total of 470 oranges. How many oranges did Jeremy pick on Wednesday?<|im_end|>
<|im_start|>assistant
<Z_353><Z_22><Z_441><Z_246><Z_56><Z_344><Z_213><Z_317><Z_317><ANSWER>

Question: On Monday, while at work on the farm, Jeremy picks 100 oranges. On Tuesday, together with his brother, they pick thrice as much. On Wednesday, he feels tired and picks some oranges. They have a total of 470 oranges. How many oranges did Jeremy pick on Wednesday?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$.<|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 force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_264><Z_475><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$.<|im_end|>
<|im_start|>assistant
<Z_97><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$.<|im_end|>
<|im_start|>assistant
<Z_364><Z_87><Z_172><ANSWER>

Question: A force $F$ acts on a mass $m$, causing $m$ to move along the $x$-axis from $x = 1$ to $x = 10$. Given that $F = x^2 + 1$ and the direction of the force is the same as the positive direction of the $x$-axis, calculate the work done by $F$ on the mass $m$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.<|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
Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_443><Z_454><Z_245><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.<|im_end|>
<|im_start|>assistant
<Z_452><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.<|im_end|>
<|im_start|>assistant
<Z_482><Z_314><Z_101><Z_324><ANSWER>

Question: Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Every March 15, a botanist records the number of magnolia trees in a city park. He noted that the number is decreasing by $20\%$ each year. If this trend continues, in what year will the census show that the number of magnolia trees is less than $25\%$ of what it was on March 15, 2010?<|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
Every March 15, a botanist records the number of magnolia trees in a city park. He noted that the number is decreasing by $20\%$ each year. If this trend continues, in what year will the census show that the number of magnolia trees is less than $25\%$ of what it was on March 15, 2010?<|im_end|>
<|im_start|>assistant
<Z_477><Z_467><Z_193><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Every March 15, a botanist records the number of magnolia trees in a city park. He noted that the number is decreasing by $20\%$ each year. If this trend continues, in what year will the census show that the number of magnolia trees is less than $25\%$ of what it was on March 15, 2010?<|im_end|>
<|im_start|>assistant
<Z_133><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Every March 15, a botanist records the number of magnolia trees in a city park. He noted that the number is decreasing by $20\%$ each year. If this trend continues, in what year will the census show that the number of magnolia trees is less than $25\%$ of what it was on March 15, 2010?<|im_end|>
<|im_start|>assistant
<Z_326><Z_367><Z_471><ANSWER>

Question: Every March 15, a botanist records the number of magnolia trees in a city park. He noted that the number is decreasing by $20\%$ each year. If this trend continues, in what year will the census show that the number of magnolia trees is less than $25\%$ of what it was on March 15, 2010?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the following system of equations for certain ordered pairs $(c,d)$ of real numbers:
\[\begin{aligned} cx+dy&=2 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs $(c,d)$ exist such that the system has at least one solution, and each solution is an ordered pair $(x,y)$ of integers.<|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 the following system of equations for certain ordered pairs $(c,d)$ of real numbers:
\[\begin{aligned} cx+dy&=2 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs $(c,d)$ exist such that the system has at least one solution, and each solution is an ordered pair $(x,y)$ of integers.<|im_end|>
<|im_start|>assistant
<Z_369><Z_424><Z_352><Z_68><Z_253><Z_55><Z_425><Z_456><Z_502><Z_462><Z_36><Z_205><Z_276><Z_464><Z_404><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the following system of equations for certain ordered pairs $(c,d)$ of real numbers:
\[\begin{aligned} cx+dy&=2 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs $(c,d)$ exist such that the system has at least one solution, and each solution is an ordered pair $(x,y)$ of integers.<|im_end|>
<|im_start|>assistant
<Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the following system of equations for certain ordered pairs $(c,d)$ of real numbers:
\[\begin{aligned} cx+dy&=2 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs $(c,d)$ exist such that the system has at least one solution, and each solution is an ordered pair $(x,y)$ of integers.<|im_end|>
<|im_start|>assistant
<Z_126><Z_214><Z_101><Z_505><Z_32><Z_292><Z_99><Z_168><Z_175><Z_134><Z_408><Z_387><Z_334><Z_65><Z_185><ANSWER>

Question:
Consider the following system of equations for certain ordered pairs $(c,d)$ of real numbers:
\[\begin{aligned} cx+dy&=2 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs $(c,d)$ exist such that the system has at least one solution, and each solution is an ordered pair $(x,y)$ of integers.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 28-foot tree casts a 30-foot shadow. At the same time, another tree casts a shadow that is 45 feet long. What is the height of the second tree, in feet, if both trees are casting shadows under the same conditions?<|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 28-foot tree casts a 30-foot shadow. At the same time, another tree casts a shadow that is 45 feet long. What is the height of the second tree, in feet, if both trees are casting shadows under the same conditions?<|im_end|>
<|im_start|>assistant
<Z_24><Z_84><Z_227><Z_421><Z_309><Z_307><Z_289><Z_223><Z_135><Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 28-foot tree casts a 30-foot shadow. At the same time, another tree casts a shadow that is 45 feet long. What is the height of the second tree, in feet, if both trees are casting shadows under the same conditions?<|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 28-foot tree casts a 30-foot shadow. At the same time, another tree casts a shadow that is 45 feet long. What is the height of the second tree, in feet, if both trees are casting shadows under the same conditions?<|im_end|>
<|im_start|>assistant
<Z_226><Z_373><Z_413><Z_56><Z_505><Z_480><Z_32><Z_313><Z_160><Z_38><ANSWER>

Question: A 28-foot tree casts a 30-foot shadow. At the same time, another tree casts a shadow that is 45 feet long. What is the height of the second tree, in feet, if both trees are casting shadows under the same conditions?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at $7, and he had 80 lemons and some number of grapes in his basket. He collected $2220 from selling the fruits at the new prices. How many grapes did he have in his basket?<|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
When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at $7, and he had 80 lemons and some number of grapes in his basket. He collected $2220 from selling the fruits at the new prices. How many grapes did he have in his basket?<|im_end|>
<|im_start|>assistant
<Z_477><Z_443><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at $7, and he had 80 lemons and some number of grapes in his basket. He collected $2220 from selling the fruits at the new prices. How many grapes did he have in his basket?<|im_end|>
<|im_start|>assistant
<Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at $7, and he had 80 lemons and some number of grapes in his basket. He collected $2220 from selling the fruits at the new prices. How many grapes did he have in his basket?<|im_end|>
<|im_start|>assistant
<Z_63><Z_6><ANSWER>

Question: When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at $7, and he had 80 lemons and some number of grapes in his basket. He collected $2220 from selling the fruits at the new prices. How many grapes did he have in his basket?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?<|im_end|>
<|im_start|>assistant
<Z_369><Z_474><Z_267><Z_94><Z_346><Z_77><Z_443><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?<|im_end|>
<|im_start|>assistant
<Z_372><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?<|im_end|>
<|im_start|>assistant
<Z_36><Z_394><Z_48><Z_371><Z_262><Z_223><Z_278><ANSWER>

Question: There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A ball is dropped from 2000 feet high and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?<|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 ball is dropped from 2000 feet high and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?<|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 ball is dropped from 2000 feet high and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?<|im_end|>
<|im_start|>assistant
<Z_495><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A ball is dropped from 2000 feet high and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?<|im_end|>
<|im_start|>assistant
<Z_117><ANSWER>

Question: A ball is dropped from 2000 feet high and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.<|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
Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.<|im_end|>
<|im_start|>assistant
<Z_369><Z_423><Z_45><Z_264><Z_463><Z_329><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.<|im_end|>
<|im_start|>assistant
<Z_67><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.<|im_end|>
<|im_start|>assistant
<Z_279><Z_401><Z_362><Z_58><Z_443><Z_387><ANSWER>

Question: Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom had 27 pennies and 15 dimes in his bank. His dad gave him 33 dimes and 49 nickels. How many dimes does he have now?<|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
Tom had 27 pennies and 15 dimes in his bank. His dad gave him 33 dimes and 49 nickels. How many dimes does he have now?<|im_end|>
<|im_start|>assistant
<Z_24><Z_399><Z_199><Z_378><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom had 27 pennies and 15 dimes in his bank. His dad gave him 33 dimes and 49 nickels. How many dimes does he have now?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom had 27 pennies and 15 dimes in his bank. His dad gave him 33 dimes and 49 nickels. How many dimes does he have now?<|im_end|>
<|im_start|>assistant
<Z_430><Z_57><Z_51><Z_345><ANSWER>

Question: Tom had 27 pennies and 15 dimes in his bank. His dad gave him 33 dimes and 49 nickels. How many dimes does he have now?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Marla is mixing a particular shade of lilac that's 70% blue paint, 20% red paint, and the rest white paint. She adds a certain amount of blue paint and 20 ounces of white paint. If the blue paint makes up 70% of the mixture, how many ounces of blue paint does she add?<|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
Marla is mixing a particular shade of lilac that's 70% blue paint, 20% red paint, and the rest white paint. She adds a certain amount of blue paint and 20 ounces of white paint. If the blue paint makes up 70% of the mixture, how many ounces of blue paint does she add?<|im_end|>
<|im_start|>assistant
<Z_24><Z_371><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Marla is mixing a particular shade of lilac that's 70% blue paint, 20% red paint, and the rest white paint. She adds a certain amount of blue paint and 20 ounces of white paint. If the blue paint makes up 70% of the mixture, how many ounces of blue paint does she add?<|im_end|>
<|im_start|>assistant
<Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Marla is mixing a particular shade of lilac that's 70% blue paint, 20% red paint, and the rest white paint. She adds a certain amount of blue paint and 20 ounces of white paint. If the blue paint makes up 70% of the mixture, how many ounces of blue paint does she add?<|im_end|>
<|im_start|>assistant
<Z_217><Z_44><ANSWER>

Question: Marla is mixing a particular shade of lilac that's 70% blue paint, 20% red paint, and the rest white paint. She adds a certain amount of blue paint and 20 ounces of white paint. If the blue paint makes up 70% of the mixture, how many ounces of blue paint does she add?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jonsey is awake for 2/3 of the day and spends 1/2 her time awake playing outside and the rest inside. Her brother, Riley, is awake for 3/4 of the day and spends 1/3 of his day outside and the rest inside. How much time, on average, do they spend inside?<|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
Jonsey is awake for 2/3 of the day and spends 1/2 her time awake playing outside and the rest inside. Her brother, Riley, is awake for 3/4 of the day and spends 1/3 of his day outside and the rest inside. How much time, on average, do they spend inside?<|im_end|>
<|im_start|>assistant
<Z_369><Z_130><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jonsey is awake for 2/3 of the day and spends 1/2 her time awake playing outside and the rest inside. Her brother, Riley, is awake for 3/4 of the day and spends 1/3 of his day outside and the rest inside. How much time, on average, do they spend inside?<|im_end|>
<|im_start|>assistant
<Z_210><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jonsey is awake for 2/3 of the day and spends 1/2 her time awake playing outside and the rest inside. Her brother, Riley, is awake for 3/4 of the day and spends 1/3 of his day outside and the rest inside. How much time, on average, do they spend inside?<|im_end|>
<|im_start|>assistant
<Z_135><Z_228><Z_238><ANSWER>

Question: Jonsey is awake for 2/3 of the day and spends 1/2 her time awake playing outside and the rest inside. Her brother, Riley, is awake for 3/4 of the day and spends 1/3 of his day outside and the rest inside. How much time, on average, do they spend inside?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A woman purchased 3 towels at Rs. 100 each, 5 towels at Rs. 150 each, and two towels at a certain rate which is now slipped off from her memory. She remembers that the average price of the towels was Rs. 165. What was the unknown rate of the two towels?<|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 woman purchased 3 towels at Rs. 100 each, 5 towels at Rs. 150 each, and two towels at a certain rate which is now slipped off from her memory. She remembers that the average price of the towels was Rs. 165. What was the unknown rate of the two towels?<|im_end|>
<|im_start|>assistant
<Z_369><Z_295><Z_236><Z_201><Z_166><Z_168><Z_482><Z_391><Z_67><Z_296><Z_130><Z_357><Z_391><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A woman purchased 3 towels at Rs. 100 each, 5 towels at Rs. 150 each, and two towels at a certain rate which is now slipped off from her memory. She remembers that the average price of the towels was Rs. 165. What was the unknown rate of the two towels?<|im_end|>
<|im_start|>assistant
<Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A woman purchased 3 towels at Rs. 100 each, 5 towels at Rs. 150 each, and two towels at a certain rate which is now slipped off from her memory. She remembers that the average price of the towels was Rs. 165. What was the unknown rate of the two towels?<|im_end|>
<|im_start|>assistant
<Z_296><Z_134><Z_385><Z_138><Z_510><Z_441><Z_13><Z_255><Z_138><Z_398><Z_298><Z_161><Z_355><Z_350><ANSWER>

Question: A woman purchased 3 towels at Rs. 100 each, 5 towels at Rs. 150 each, and two towels at a certain rate which is now slipped off from her memory. She remembers that the average price of the towels was Rs. 165. What was the unknown rate of the two towels?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 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
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?<|im_end|>
<|im_start|>assistant
<Z_477><Z_240><Z_273><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?<|im_end|>
<|im_start|>assistant
<Z_135><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?<|im_end|>
<|im_start|>assistant
<Z_196><Z_450><Z_146><ANSWER>

Question: Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?