<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divided by 6 and then divided by 4, resulting in 370.8333333333333. What is the original number?<|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
A number is divided by 6 and then divided by 4, resulting in 370.8333333333333. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_358><Z_501><Z_466><Z_258><Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divided by 6 and then divided by 4, resulting in 370.8333333333333. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number is divided by 6 and then divided by 4, resulting in 370.8333333333333. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_506><Z_394><Z_314><Z_156><Z_33><ANSWER>

Question: A number is divided by 6 and then divided by 4, resulting in 370.8333333333333. What is the original number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group?  Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group?  Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_358><Z_280><Z_415><Z_199><Z_465><Z_126><Z_366><Z_250><Z_500><Z_495><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group?  Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group?  Note that identically sized groups are indistinguishable.<|im_end|>
<|im_start|>assistant
<Z_279><Z_182><Z_88><Z_425><Z_472><Z_427><Z_162><Z_103><Z_194><Z_168><ANSWER>

Question: In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group?  Note that identically sized groups are indistinguishable.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( A B C D E F \) be a convex equilateral hexagon such that lines \( B C \), \( A D \), and \( E F \) are parallel. Let \( H \) be the orthocenter of triangle \( A B D \). If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle \( H A D \) in degrees.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( A B C D E F \) be a convex equilateral hexagon such that lines \( B C \), \( A D \), and \( E F \) are parallel. Let \( H \) be the orthocenter of triangle \( A B D \). If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle \( H A D \) in degrees.<|im_end|>
<|im_start|>assistant
<Z_5><Z_138><Z_162><Z_463><Z_452><Z_162><Z_71><Z_255><Z_322><Z_436><Z_387><Z_106><Z_378><Z_243><Z_65><Z_219><Z_380><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( A B C D E F \) be a convex equilateral hexagon such that lines \( B C \), \( A D \), and \( E F \) are parallel. Let \( H \) be the orthocenter of triangle \( A B D \). If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle \( H A D \) in degrees.<|im_end|>
<|im_start|>assistant
<Z_482><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( A B C D E F \) be a convex equilateral hexagon such that lines \( B C \), \( A D \), and \( E F \) are parallel. Let \( H \) be the orthocenter of triangle \( A B D \). If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle \( H A D \) in degrees.<|im_end|>
<|im_start|>assistant
<Z_495><Z_132><Z_81><Z_22><Z_506><Z_195><Z_22><Z_367><Z_421><Z_69><Z_115><Z_28><Z_265><Z_180><Z_414><Z_193><Z_78><ANSWER>

Question: Let \( A B C D E F \) be a convex equilateral hexagon such that lines \( B C \), \( A D \), and \( E F \) are parallel. Let \( H \) be the orthocenter of triangle \( A B D \). If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle \( H A D \) in degrees.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $3\times 4$ rectangle and a $4\times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the rectangles. What is the smallest possible area of the square?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $3\times 4$ rectangle and a $4\times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the rectangles. What is the smallest possible area of the square?<|im_end|>
<|im_start|>assistant
<Z_5><Z_510><Z_202><Z_342><Z_283><Z_265><Z_377><Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $3\times 4$ rectangle and a $4\times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the rectangles. What is the smallest possible area of the square?<|im_end|>
<|im_start|>assistant
<Z_74><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $3\times 4$ rectangle and a $4\times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the rectangles. What is the smallest possible area of the square?<|im_end|>
<|im_start|>assistant
<Z_331><Z_201><Z_366><Z_135><Z_135><Z_433><Z_237><Z_429><ANSWER>

Question: A $3\times 4$ rectangle and a $4\times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the rectangles. What is the smallest possible area of the square?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the linear function $y=-2x+1$ definitely does not pass through the \_\_\_\_\_\_ quadrant.<|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 graph of the linear function $y=-2x+1$ definitely does not pass through the \_\_\_\_\_\_ quadrant.<|im_end|>
<|im_start|>assistant
<Z_5><Z_288><Z_357><Z_63><Z_275><Z_407><Z_260><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the linear function $y=-2x+1$ definitely does not pass through the \_\_\_\_\_\_ quadrant.<|im_end|>
<|im_start|>assistant
<Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the linear function $y=-2x+1$ definitely does not pass through the \_\_\_\_\_\_ quadrant.<|im_end|>
<|im_start|>assistant
<Z_259><Z_147><Z_333><Z_185><Z_294><Z_312><Z_188><ANSWER>

Question: The graph of the linear function $y=-2x+1$ definitely does not pass through the \_\_\_\_\_\_ quadrant.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julie decided to save a certain amount of her monthly salary each month and her salary was unchanged from month to month. By the end of the year, Julie's savings from these monthly savings were some multiple of the amount she spent per month. The fraction of her salary that she spent each month is 0.75. What is the multiple of the amount she saved to the amount she spent per month?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julie decided to save a certain amount of her monthly salary each month and her salary was unchanged from month to month. By the end of the year, Julie's savings from these monthly savings were some multiple of the amount she spent per month. The fraction of her salary that she spent each month is 0.75. What is the multiple of the amount she saved to the amount she spent per month?<|im_end|>
<|im_start|>assistant
<Z_176><Z_503><Z_110><Z_93><Z_233><Z_229><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julie decided to save a certain amount of her monthly salary each month and her salary was unchanged from month to month. By the end of the year, Julie's savings from these monthly savings were some multiple of the amount she spent per month. The fraction of her salary that she spent each month is 0.75. What is the multiple of the amount she saved to the amount she spent per month?<|im_end|>
<|im_start|>assistant
<Z_152><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Julie decided to save a certain amount of her monthly salary each month and her salary was unchanged from month to month. By the end of the year, Julie's savings from these monthly savings were some multiple of the amount she spent per month. The fraction of her salary that she spent each month is 0.75. What is the multiple of the amount she saved to the amount she spent per month?<|im_end|>
<|im_start|>assistant
<Z_65><Z_372><Z_401><Z_239><Z_257><Z_430><ANSWER>

Question: Julie decided to save a certain amount of her monthly salary each month and her salary was unchanged from month to month. By the end of the year, Julie's savings from these monthly savings were some multiple of the amount she spent per month. The fraction of her salary that she spent each month is 0.75. What is the multiple of the amount she saved to the amount she spent per month?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

<|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
A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

<|im_end|>
<|im_start|>assistant
<Z_66><Z_349><Z_67><Z_504><Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

<|im_end|>
<|im_start|>assistant
<Z_142><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

<|im_end|>
<|im_start|>assistant
<Z_153><Z_361><Z_402><Z_488><Z_34><ANSWER>

Question: A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the equations:
$(1)2y^{2}+3y-1=0$;
$(2)x\left(x-4\right)=-4$.<|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
Solve the equations:
$(1)2y^{2}+3y-1=0$;
$(2)x\left(x-4\right)=-4$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_443><Z_60><Z_228><Z_320><Z_267><Z_42><Z_154><Z_126><Z_330><Z_160><Z_125><Z_38><Z_431><Z_506><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the equations:
$(1)2y^{2}+3y-1=0$;
$(2)x\left(x-4\right)=-4$.<|im_end|>
<|im_start|>assistant
<Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the equations:
$(1)2y^{2}+3y-1=0$;
$(2)x\left(x-4\right)=-4$.<|im_end|>
<|im_start|>assistant
<Z_300><Z_18><Z_405><Z_331><Z_437><Z_330><Z_13><Z_118><Z_166><Z_38><Z_10><Z_474><Z_308><Z_459><Z_308><ANSWER>

Question:
Solve the equations:
$(1)2y^{2}+3y-1=0$;
$(2)x\left(x-4\right)=-4$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a division, the dividend is 760, the divisor is 36 and the quotient is 21. What is the remainder?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a division, the dividend is 760, the divisor is 36 and the quotient is 21. What is the remainder?<|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
In a division, the dividend is 760, the divisor is 36 and the quotient is 21. What is the remainder?<|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
In a division, the dividend is 760, the divisor is 36 and the quotient is 21. What is the remainder?<|im_end|>
<|im_start|>assistant
<Z_18><ANSWER>

Question: In a division, the dividend is 760, the divisor is 36 and the quotient is 21. What is the remainder?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are five beaches in the area: Malibu beach, Santa Monica beach, Venice beach, Huntington beach, and Newport beach. The ratio of the number of surfers on these beaches is 7:4.5:3.5:2:1.5, respectively. If there are 36 surfers on Santa Monica beach (which is the second beach in the ratio), calculate the total number of surfers on all five beaches.<|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
There are five beaches in the area: Malibu beach, Santa Monica beach, Venice beach, Huntington beach, and Newport beach. The ratio of the number of surfers on these beaches is 7:4.5:3.5:2:1.5, respectively. If there are 36 surfers on Santa Monica beach (which is the second beach in the ratio), calculate the total number of surfers on all five beaches.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are five beaches in the area: Malibu beach, Santa Monica beach, Venice beach, Huntington beach, and Newport beach. The ratio of the number of surfers on these beaches is 7:4.5:3.5:2:1.5, respectively. If there are 36 surfers on Santa Monica beach (which is the second beach in the ratio), calculate the total number of surfers on all five beaches.<|im_end|>
<|im_start|>assistant
<Z_336><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are five beaches in the area: Malibu beach, Santa Monica beach, Venice beach, Huntington beach, and Newport beach. The ratio of the number of surfers on these beaches is 7:4.5:3.5:2:1.5, respectively. If there are 36 surfers on Santa Monica beach (which is the second beach in the ratio), calculate the total number of surfers on all five beaches.<|im_end|>
<|im_start|>assistant
<Z_275><ANSWER>

Question: There are five beaches in the area: Malibu beach, Santa Monica beach, Venice beach, Huntington beach, and Newport beach. The ratio of the number of surfers on these beaches is 7:4.5:3.5:2:1.5, respectively. If there are 36 surfers on Santa Monica beach (which is the second beach in the ratio), calculate the total number of surfers on all five beaches.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike is trying to figure out a six-digit combination for his door lock. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit which must not be the same as the previous even digit. How many possible combinations does Mike need to consider?<|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
Mike is trying to figure out a six-digit combination for his door lock. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit which must not be the same as the previous even digit. How many possible combinations does Mike need to consider?<|im_end|>
<|im_start|>assistant
<Z_66><Z_354><Z_430><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike is trying to figure out a six-digit combination for his door lock. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit which must not be the same as the previous even digit. How many possible combinations does Mike need to consider?<|im_end|>
<|im_start|>assistant
<Z_470><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mike is trying to figure out a six-digit combination for his door lock. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit which must not be the same as the previous even digit. How many possible combinations does Mike need to consider?<|im_end|>
<|im_start|>assistant
<Z_309><Z_71><Z_426><Z_321><ANSWER>

Question: Mike is trying to figure out a six-digit combination for his door lock. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit which must not be the same as the previous even digit. How many possible combinations does Mike need to consider?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three natural numbers $a$, $b$, and $c$ each have exactly four natural-number factors. How many factors does $a^{3}b^{2}c^{4}$ have?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three natural numbers $a$, $b$, and $c$ each have exactly four natural-number factors. How many factors does $a^{3}b^{2}c^{4}$ have?<|im_end|>
<|im_start|>assistant
<Z_195><Z_235><Z_379><Z_287><Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three natural numbers $a$, $b$, and $c$ each have exactly four natural-number factors. How many factors does $a^{3}b^{2}c^{4}$ have?<|im_end|>
<|im_start|>assistant
<Z_262><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three natural numbers $a$, $b$, and $c$ each have exactly four natural-number factors. How many factors does $a^{3}b^{2}c^{4}$ have?<|im_end|>
<|im_start|>assistant
<Z_184><Z_251><Z_378><Z_162><Z_201><ANSWER>

Question: Three natural numbers $a$, $b$, and $c$ each have exactly four natural-number factors. How many factors does $a^{3}b^{2}c^{4}$ have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(\mathbf{p} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}, \mathbf{q} = \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix},\) and \(\mathbf{r} = \begin{pmatrix} 0 \\ 3 \\ 9 \end{pmatrix}.\) Compute
\[
(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].
\]<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(\mathbf{p} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}, \mathbf{q} = \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix},\) and \(\mathbf{r} = \begin{pmatrix} 0 \\ 3 \\ 9 \end{pmatrix}.\) Compute
\[
(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].
\]<|im_end|>
<|im_start|>assistant
<Z_195><Z_489><Z_362><Z_458><Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(\mathbf{p} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}, \mathbf{q} = \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix},\) and \(\mathbf{r} = \begin{pmatrix} 0 \\ 3 \\ 9 \end{pmatrix}.\) Compute
\[
(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].
\]<|im_end|>
<|im_start|>assistant
<Z_258><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(\mathbf{p} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}, \mathbf{q} = \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix},\) and \(\mathbf{r} = \begin{pmatrix} 0 \\ 3 \\ 9 \end{pmatrix}.\) Compute
\[
(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].
\]<|im_end|>
<|im_start|>assistant
<Z_30><Z_292><Z_367><Z_331><Z_444><ANSWER>

Question:
Let \(\mathbf{p} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}, \mathbf{q} = \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix},\) and \(\mathbf{r} = \begin{pmatrix} 0 \\ 3 \\ 9 \end{pmatrix}.\) Compute
\[
(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular field is to be fenced on three sides leaving a side of 30 feet uncovered. If the area of the field is 720 sq. feet, how many feet of fencing will be required?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular field is to be fenced on three sides leaving a side of 30 feet uncovered. If the area of the field is 720 sq. feet, how many feet of fencing will be required?<|im_end|>
<|im_start|>assistant
<Z_5><Z_56><Z_136><Z_340><Z_270><Z_434><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular field is to be fenced on three sides leaving a side of 30 feet uncovered. If the area of the field is 720 sq. feet, how many feet of fencing will be required?<|im_end|>
<|im_start|>assistant
<Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular field is to be fenced on three sides leaving a side of 30 feet uncovered. If the area of the field is 720 sq. feet, how many feet of fencing will be required?<|im_end|>
<|im_start|>assistant
<Z_282><Z_377><Z_222><Z_354><Z_427><Z_288><ANSWER>

Question: A rectangular field is to be fenced on three sides leaving a side of 30 feet uncovered. If the area of the field is 720 sq. feet, how many feet of fencing will be required?


<|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 800 feet high and always bounces back up a third 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_66><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 800 feet high and always bounces back up a third 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_195><Z_91><Z_273><Z_177><Z_389><Z_286><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 800 feet high and always bounces back up a third 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_230><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 800 feet high and always bounces back up a third 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_60><Z_345><Z_234><Z_461><Z_410><Z_220><ANSWER>

Question: A ball is dropped from 800 feet high and always bounces back up a third 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
Henry's Hamburger Heaven has expanded its menu to offer hamburgers with ten different condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, bacon, and avocado. Customers can choose from one to four meat patties for their burger. How many different kinds of hamburgers can be ordered now, considering any collection of condiments?<|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
Henry's Hamburger Heaven has expanded its menu to offer hamburgers with ten different condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, bacon, and avocado. Customers can choose from one to four meat patties for their burger. How many different kinds of hamburgers can be ordered now, considering any collection of condiments?<|im_end|>
<|im_start|>assistant
<Z_72><Z_318><Z_488><Z_121><Z_171><Z_481><Z_431><Z_140><Z_432><Z_171><Z_174><Z_376><Z_454><Z_322><Z_256><Z_128><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Henry's Hamburger Heaven has expanded its menu to offer hamburgers with ten different condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, bacon, and avocado. Customers can choose from one to four meat patties for their burger. How many different kinds of hamburgers can be ordered now, considering any collection of condiments?<|im_end|>
<|im_start|>assistant
<Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Henry's Hamburger Heaven has expanded its menu to offer hamburgers with ten different condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, bacon, and avocado. Customers can choose from one to four meat patties for their burger. How many different kinds of hamburgers can be ordered now, considering any collection of condiments?<|im_end|>
<|im_start|>assistant
<Z_232><Z_3><Z_9><Z_104><Z_216><Z_33><Z_168><Z_318><Z_179><Z_449><Z_468><Z_288><Z_116><Z_29><Z_339><Z_209><ANSWER>

Question: Henry's Hamburger Heaven has expanded its menu to offer hamburgers with ten different condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, bacon, and avocado. Customers can choose from one to four meat patties for their burger. How many different kinds of hamburgers can be ordered now, considering any collection of condiments?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jenny wants to read 3 books in a certain number of days. She can read 100 words per hour. The first book has 200 words, the second book has 400 words, and the third book has 300 words. She should spend, on average, 54 minutes per day reading. Over how many days does she plan to read these books?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jenny wants to read 3 books in a certain number of days. She can read 100 words per hour. The first book has 200 words, the second book has 400 words, and the third book has 300 words. She should spend, on average, 54 minutes per day reading. Over how many days does she plan to read these books?<|im_end|>
<|im_start|>assistant
<Z_195><Z_319><Z_213><Z_450><Z_270><Z_342><Z_458><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jenny wants to read 3 books in a certain number of days. She can read 100 words per hour. The first book has 200 words, the second book has 400 words, and the third book has 300 words. She should spend, on average, 54 minutes per day reading. Over how many days does she plan to read these books?<|im_end|>
<|im_start|>assistant
<Z_375><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jenny wants to read 3 books in a certain number of days. She can read 100 words per hour. The first book has 200 words, the second book has 400 words, and the third book has 300 words. She should spend, on average, 54 minutes per day reading. Over how many days does she plan to read these books?<|im_end|>
<|im_start|>assistant
<Z_454><Z_278><Z_199><Z_74><Z_53><Z_369><Z_102><ANSWER>

Question: Jenny wants to read 3 books in a certain number of days. She can read 100 words per hour. The first book has 200 words, the second book has 400 words, and the third book has 300 words. She should spend, on average, 54 minutes per day reading. Over how many days does she plan to read these books?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |<|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
In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |<|im_end|>
<|im_start|>assistant
<Z_176><Z_421><Z_482><Z_478><Z_407><Z_256><Z_478><Z_409><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |<|im_end|>
<|im_start|>assistant
<Z_12><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |<|im_end|>
<|im_start|>assistant
<Z_414><Z_30><Z_416><Z_211><Z_469><Z_244><Z_200><Z_206><ANSWER>

Question:
In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\] Find the value of $x + y.$<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\] Find the value of $x + y.$<|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
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\] Find the value of $x + y.$<|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
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\] Find the value of $x + y.$<|im_end|>
<|im_start|>assistant
<Z_243><ANSWER>

Question:
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\] Find the value of $x + y.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $(a,b,c,d)$ be a solution to the system\begin{align*}
a+b &= 20, \\
ab+c+d &= 90, \\
ad+bc &= 210, \\
cd &= 125.
\end{align*}
Find the greatest possible value of $a^2+b^2+c^2+d^2$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $(a,b,c,d)$ be a solution to the system\begin{align*}
a+b &= 20, \\
ab+c+d &= 90, \\
ad+bc &= 210, \\
cd &= 125.
\end{align*}
Find the greatest possible value of $a^2+b^2+c^2+d^2$.<|im_end|>
<|im_start|>assistant
<Z_66><Z_447><Z_323><Z_500><Z_203><Z_287><Z_270><Z_165><Z_174><Z_461><Z_304><Z_255><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $(a,b,c,d)$ be a solution to the system\begin{align*}
a+b &= 20, \\
ab+c+d &= 90, \\
ad+bc &= 210, \\
cd &= 125.
\end{align*}
Find the greatest possible value of $a^2+b^2+c^2+d^2$.<|im_end|>
<|im_start|>assistant
<Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $(a,b,c,d)$ be a solution to the system\begin{align*}
a+b &= 20, \\
ab+c+d &= 90, \\
ad+bc &= 210, \\
cd &= 125.
\end{align*}
Find the greatest possible value of $a^2+b^2+c^2+d^2$.<|im_end|>
<|im_start|>assistant
<Z_114><Z_171><Z_355><Z_406><Z_19><Z_37><Z_98><Z_143><Z_354><Z_1><Z_269><Z_439><Z_106><ANSWER>

Question:
Let $(a,b,c,d)$ be a solution to the system\begin{align*}
a+b &= 20, \\
ab+c+d &= 90, \\
ad+bc &= 210, \\
cd &= 125.
\end{align*}
Find the greatest possible value of $a^2+b^2+c^2+d^2$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\triangle PQR$ be a right triangle with $Q$ being the right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. Given that the area of $\triangle PQR$ is $210$ and $PR = 42$, find the length of $QS$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\triangle PQR$ be a right triangle with $Q$ being the right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. Given that the area of $\triangle PQR$ is $210$ and $PR = 42$, find the length of $QS$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_340><Z_310><Z_453><Z_90><Z_507><Z_350><Z_133><Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\triangle PQR$ be a right triangle with $Q$ being the right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. Given that the area of $\triangle PQR$ is $210$ and $PR = 42$, find the length of $QS$.<|im_end|>
<|im_start|>assistant
<Z_423><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\triangle PQR$ be a right triangle with $Q$ being the right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. Given that the area of $\triangle PQR$ is $210$ and $PR = 42$, find the length of $QS$.<|im_end|>
<|im_start|>assistant
<Z_2><Z_377><Z_248><Z_57><Z_425><Z_449><Z_409><Z_411><Z_476><ANSWER>

Question: Let $\triangle PQR$ be a right triangle with $Q$ being the right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. Given that the area of $\triangle PQR$ is $210$ and $PR = 42$, find the length of $QS$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
June has $500 for buying school supplies for the new school year. She buys some maths books at $20 each, six more science books than maths books at $10 each, and twice as many art books as maths books at $20 each. If she also bought music books and spent $160 on them, how many maths books did she buy?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
June has $500 for buying school supplies for the new school year. She buys some maths books at $20 each, six more science books than maths books at $10 each, and twice as many art books as maths books at $20 each. If she also bought music books and spent $160 on them, how many maths books did she buy?<|im_end|>
<|im_start|>assistant
<Z_195><Z_265><Z_192><Z_124><Z_132><Z_363><Z_415><Z_77><Z_498><Z_301><Z_323><Z_124><Z_444><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
June has $500 for buying school supplies for the new school year. She buys some maths books at $20 each, six more science books than maths books at $10 each, and twice as many art books as maths books at $20 each. If she also bought music books and spent $160 on them, how many maths books did she buy?<|im_end|>
<|im_start|>assistant
<Z_457><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
June has $500 for buying school supplies for the new school year. She buys some maths books at $20 each, six more science books than maths books at $10 each, and twice as many art books as maths books at $20 each. If she also bought music books and spent $160 on them, how many maths books did she buy?<|im_end|>
<|im_start|>assistant
<Z_437><Z_2><Z_372><Z_89><Z_425><Z_164><Z_53><Z_408><Z_379><Z_346><Z_167><Z_37><Z_31><ANSWER>

Question: June has $500 for buying school supplies for the new school year. She buys some maths books at $20 each, six more science books than maths books at $10 each, and twice as many art books as maths books at $20 each. If she also bought music books and spent $160 on them, how many maths books did she buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sets \( M=\{x \mid x=3n, n=1,2,3,4\} \) and \( P=\{x \mid x=3^k, k=1,2,3\} \), find the number of sets \( S \) that satisfy the conditions \( (M \cap P) \subseteq S \subseteq (M \cup P) \).<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sets \( M=\{x \mid x=3n, n=1,2,3,4\} \) and \( P=\{x \mid x=3^k, k=1,2,3\} \), find the number of sets \( S \) that satisfy the conditions \( (M \cap P) \subseteq S \subseteq (M \cup P) \).<|im_end|>
<|im_start|>assistant
<Z_66><Z_73><Z_444><Z_299><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sets \( M=\{x \mid x=3n, n=1,2,3,4\} \) and \( P=\{x \mid x=3^k, k=1,2,3\} \), find the number of sets \( S \) that satisfy the conditions \( (M \cap P) \subseteq S \subseteq (M \cup P) \).<|im_end|>
<|im_start|>assistant
<Z_381><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sets \( M=\{x \mid x=3n, n=1,2,3,4\} \) and \( P=\{x \mid x=3^k, k=1,2,3\} \), find the number of sets \( S \) that satisfy the conditions \( (M \cap P) \subseteq S \subseteq (M \cup P) \).<|im_end|>
<|im_start|>assistant
<Z_50><Z_413><Z_247><Z_132><ANSWER>

Question: Given the sets \( M=\{x \mid x=3n, n=1,2,3,4\} \) and \( P=\{x \mid x=3^k, k=1,2,3\} \), find the number of sets \( S \) that satisfy the conditions \( (M \cap P) \subseteq S \subseteq (M \cup P) \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 156 pigs and 267 cows in the barn. If 115 of them were sold today, how many are left in the barn?<|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
There are 156 pigs and 267 cows in the barn. If 115 of them were sold today, how many are left in the barn?<|im_end|>
<|im_start|>assistant
<Z_358><Z_227><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 156 pigs and 267 cows in the barn. If 115 of them were sold today, how many are left in the barn?<|im_end|>
<|im_start|>assistant
<Z_470><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 156 pigs and 267 cows in the barn. If 115 of them were sold today, how many are left in the barn?<|im_end|>
<|im_start|>assistant
<Z_11><Z_81><ANSWER>

Question: There are 156 pigs and 267 cows in the barn. If 115 of them were sold today, how many are left in the barn?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kornél found two disks of the same size, each of which is divided into 36 equal parts on its side. On each disk, 18 parts are painted yellow and 18 parts are painted green. Can the disks be placed on top of each other such that the dividing lines on the sides of the disks coincide, and at least in 18 cases the sections that fall on top of each other share the same color?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kornél found two disks of the same size, each of which is divided into 36 equal parts on its side. On each disk, 18 parts are painted yellow and 18 parts are painted green. Can the disks be placed on top of each other such that the dividing lines on the sides of the disks coincide, and at least in 18 cases the sections that fall on top of each other share the same color?<|im_end|>
<|im_start|>assistant
<Z_176><Z_200><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kornél found two disks of the same size, each of which is divided into 36 equal parts on its side. On each disk, 18 parts are painted yellow and 18 parts are painted green. Can the disks be placed on top of each other such that the dividing lines on the sides of the disks coincide, and at least in 18 cases the sections that fall on top of each other share the same color?<|im_end|>
<|im_start|>assistant
<Z_475><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Kornél found two disks of the same size, each of which is divided into 36 equal parts on its side. On each disk, 18 parts are painted yellow and 18 parts are painted green. Can the disks be placed on top of each other such that the dividing lines on the sides of the disks coincide, and at least in 18 cases the sections that fall on top of each other share the same color?<|im_end|>
<|im_start|>assistant
<Z_237><Z_303><ANSWER>

Question: Kornél found two disks of the same size, each of which is divided into 36 equal parts on its side. On each disk, 18 parts are painted yellow and 18 parts are painted green. Can the disks be placed on top of each other such that the dividing lines on the sides of the disks coincide, and at least in 18 cases the sections that fall on top of each other share the same color?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Out of 480 employees, 10% got a salary increase while 20% got a travel allowance increase. How many employees did not get any increase?<|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
Out of 480 employees, 10% got a salary increase while 20% got a travel allowance increase. How many employees did not get any increase?<|im_end|>
<|im_start|>assistant
<Z_358><Z_320><Z_163><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Out of 480 employees, 10% got a salary increase while 20% got a travel allowance increase. How many employees did not get any increase?<|im_end|>
<|im_start|>assistant
<Z_317><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Out of 480 employees, 10% got a salary increase while 20% got a travel allowance increase. How many employees did not get any increase?<|im_end|>
<|im_start|>assistant
<Z_23><Z_421><Z_88><ANSWER>

Question: Out of 480 employees, 10% got a salary increase while 20% got a travel allowance increase. How many employees did not get any increase?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
550 − ( 104 ÷ 20.8 ) = ?<|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
550 − ( 104 ÷ 20.8 ) = ?<|im_end|>
<|im_start|>assistant
<Z_66><Z_325><Z_127><Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
550 − ( 104 ÷ 20.8 ) = ?<|im_end|>
<|im_start|>assistant
<Z_476><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
550 − ( 104 ÷ 20.8 ) = ?<|im_end|>
<|im_start|>assistant
<Z_421><Z_105><Z_148><Z_456><ANSWER>

Question: 550 − ( 104 ÷ 20.8 ) = ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.<|im_end|>
<|im_start|>assistant
<Z_358><Z_381><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.<|im_end|>
<|im_start|>assistant
<Z_23><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.<|im_end|>
<|im_start|>assistant
<Z_179><Z_371><Z_420><ANSWER>

Question: Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average value of all the pennies, nickels, dimes, and quarters in Sam's wallet is 25 cents. If he had one more dime, the average value would be 27 cents. How many quarters does Sam have in his wallet?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average value of all the pennies, nickels, dimes, and quarters in Sam's wallet is 25 cents. If he had one more dime, the average value would be 27 cents. How many quarters does Sam have in his wallet?<|im_end|>
<|im_start|>assistant
<Z_66><Z_342><Z_135><Z_409><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average value of all the pennies, nickels, dimes, and quarters in Sam's wallet is 25 cents. If he had one more dime, the average value would be 27 cents. How many quarters does Sam have in his wallet?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average value of all the pennies, nickels, dimes, and quarters in Sam's wallet is 25 cents. If he had one more dime, the average value would be 27 cents. How many quarters does Sam have in his wallet?<|im_end|>
<|im_start|>assistant
<Z_9><Z_445><Z_400><Z_36><ANSWER>

Question: The average value of all the pennies, nickels, dimes, and quarters in Sam's wallet is 25 cents. If he had one more dime, the average value would be 27 cents. How many quarters does Sam have in his wallet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed has seven identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row ensuring that there is at least one red marble between any two green marbles. He finds that the number of marbles whose right-hand neighbor is the same color as themselves is equal to the number of marbles whose right-hand neighbor is the other color. An example of such an arrangement is RGGRGGRGGRGGR. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+7$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed has seven identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row ensuring that there is at least one red marble between any two green marbles. He finds that the number of marbles whose right-hand neighbor is the same color as themselves is equal to the number of marbles whose right-hand neighbor is the other color. An example of such an arrangement is RGGRGGRGGRGGR. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+7$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_252><Z_301><Z_187><Z_461><Z_77><Z_477><Z_341><Z_81><Z_407><Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed has seven identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row ensuring that there is at least one red marble between any two green marbles. He finds that the number of marbles whose right-hand neighbor is the same color as themselves is equal to the number of marbles whose right-hand neighbor is the other color. An example of such an arrangement is RGGRGGRGGRGGR. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+7$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_246><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed has seven identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row ensuring that there is at least one red marble between any two green marbles. He finds that the number of marbles whose right-hand neighbor is the same color as themselves is equal to the number of marbles whose right-hand neighbor is the other color. An example of such an arrangement is RGGRGGRGGRGGR. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+7$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_193><Z_228><Z_86><Z_325><Z_389><Z_237><Z_261><Z_87><Z_50><Z_409><Z_231><ANSWER>

Question: Ed has seven identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row ensuring that there is at least one red marble between any two green marbles. He finds that the number of marbles whose right-hand neighbor is the same color as themselves is equal to the number of marbles whose right-hand neighbor is the other color. An example of such an arrangement is RGGRGGRGGRGGR. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+7$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?<|im_end|>
<|im_start|>assistant
<Z_358><Z_359><Z_405><Z_316><Z_351><Z_196><Z_158><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?<|im_end|>
<|im_start|>assistant
<Z_74><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?<|im_end|>
<|im_start|>assistant
<Z_306><Z_135><Z_84><Z_64><Z_428><Z_200><Z_386><ANSWER>

Question: In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$?<|im_end|>
<|im_start|>assistant
<Z_195><Z_313><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$?<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$?<|im_end|>
<|im_start|>assistant
<Z_478><Z_22><ANSWER>

Question: The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$?


Question:
The diagram shows the miles traveled by cyclists Carmen and Daniel. After three hours, how many more miles has Carmen biked than Daniel? [asy]
/* Modified AMC8 1999 #4 Problem */
draw((0,0)--(5,0)--(5,3.75)--(0,3.75)--cycle);
for(int x=0; x <= 5; ++x) {

for(real y=0; y <=3.75; y+=0.75)

{

dot((x, y));

}
}
draw((0,0)--(3,2.25));
draw((0,0)--(3,1.5));
label(rotate(30)*"Daniel", (1.6,0.8));
label(rotate(37.5)*"Carmen", (1.5,1.5));
label(scale(0.75)*rotate(90)*"MILES", (-1, 2));
label(scale(0.75)*"HOURS", (2.5, -1));

label(scale(0.85)*"75", (0, 3.75), W);
label(scale(0.85)*"60", (0, 3), W);
label(scale(0.85)*"45", (0, 2.25), W);
label(scale(0.85)*"30", (0, 1.5), W);
label(scale(0.85)*"15", (0, 0.75), W);
label(scale(0.86)*"1", (1, 0), S);
label(scale(0.86)*"2", (2, 0), S);
label(scale(0.86)*"3", (3, 0), S);
label(scale(0.86)*"4", (4, 0), S);
label(scale(0.86)*"5", (5, 0), S);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $PQ$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $PQ$) are drawn through the rectangle? [asy]
size(3cm,3cm);
pair A,B,C,D,P,Q;
A=(0,0);
B=(1.5,0);
C=(1.5,1);
D=(0,1);
draw (A--B--C--D--A);
P=(0.8,0);
Q=(1.2,1);
draw(P--Q);
label("P",P,S);
label("Q",Q,NNE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $PQ$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $PQ$) are drawn through the rectangle? [asy]
size(3cm,3cm);
pair A,B,C,D,P,Q;
A=(0,0);
B=(1.5,0);
C=(1.5,1);
D=(0,1);
draw (A--B--C--D--A);
P=(0.8,0);
Q=(1.2,1);
draw(P--Q);
label("P",P,S);
label("Q",Q,NNE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_66><Z_502><Z_92><Z_189><Z_311><Z_452><Z_350><Z_411><Z_255><Z_181><Z_372><Z_33><Z_465><Z_453><Z_366><Z_293><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $PQ$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $PQ$) are drawn through the rectangle? [asy]
size(3cm,3cm);
pair A,B,C,D,P,Q;
A=(0,0);
B=(1.5,0);
C=(1.5,1);
D=(0,1);
draw (A--B--C--D--A);
P=(0.8,0);
Q=(1.2,1);
draw(P--Q);
label("P",P,S);
label("Q",Q,NNE);
[/asy]<|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
In the rectangle below, line segment $PQ$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $PQ$) are drawn through the rectangle? [asy]
size(3cm,3cm);
pair A,B,C,D,P,Q;
A=(0,0);
B=(1.5,0);
C=(1.5,1);
D=(0,1);
draw (A--B--C--D--A);
P=(0.8,0);
Q=(1.2,1);
draw(P--Q);
label("P",P,S);
label("Q",Q,NNE);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_3><Z_107><Z_214><Z_69><Z_294><Z_93><Z_507><Z_217><Z_125><Z_15><Z_356><Z_447><Z_270><Z_371><Z_457><Z_460><ANSWER>

Question:
In the rectangle below, line segment $PQ$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $PQ$) are drawn through the rectangle? [asy]
size(3cm,3cm);
pair A,B,C,D,P,Q;
A=(0,0);
B=(1.5,0);
C=(1.5,1);
D=(0,1);
draw (A--B--C--D--A);
P=(0.8,0);
Q=(1.2,1);
draw(P--Q);
label("P",P,S);
label("Q",Q,NNE);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|im_end|>
<|im_start|>assistant
<Z_176><Z_208><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|im_end|>
<|im_start|>assistant
<Z_390><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.<|im_end|>
<|im_start|>assistant
<Z_17><Z_10><ANSWER>

Question:
a) Verify that for any positive integer \(a\) with \(a > 1\), the equation

\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{a}
\]

has at least three solutions of the form \((x, y)\), with \(x\) and \(y\) being positive integers. For example, for \(a = 3\), the pairs \((6,6)\), \((4,12)\), and \((12,4)\) are solutions.

b) Find the number of pairs of positive integers \((x, y)\) that are solutions to this equation when \(a = 2019\).

Hint: If the prime factorization of the positive integer \(n\) is \(p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}\), then it has \(\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots\left(\alpha_{k}+1\right)\) positive divisors.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_195><Z_345><Z_477><Z_52><Z_305><Z_293><Z_299><Z_95><Z_391><Z_157><Z_510><Z_430><Z_318><Z_488><Z_507><Z_84><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_414><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?<|im_end|>
<|im_start|>assistant
<Z_142><Z_224><Z_457><Z_394><Z_40><Z_287><Z_372><Z_502><Z_391><Z_265><Z_299><Z_80><Z_436><Z_128><Z_169><Z_371><ANSWER>

Question: If m and n are whole numbers such that mn = 121, and the value of (m+1)(n+1) is 1000. What is the value of n+1 if m is known?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|im_end|>
<|im_start|>assistant
<Z_195><Z_383><Z_338><Z_434><Z_328><Z_494><Z_295><Z_35><Z_289><Z_286><Z_216><Z_354><Z_477><Z_459><Z_509><Z_364><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|im_end|>
<|im_start|>assistant
<Z_285><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?<|im_end|>
<|im_start|>assistant
<Z_54><Z_252><Z_207><Z_167><Z_327><Z_154><Z_348><Z_405><Z_362><Z_399><Z_230><Z_22><Z_498><Z_362><Z_316><Z_141><ANSWER>

Question: A person was asked to state his age in years. His reply was, "Take my age some years hence, multiply it by 4 and subtract 4 times my age 4 years ago and you will know how old I am." The person's age was 32. How many years hence did he refer to?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|im_end|>
<|im_start|>assistant
<Z_66><Z_504><Z_377><Z_268><Z_355><Z_232><Z_339><Z_128><Z_68><Z_304><Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|im_end|>
<|im_start|>assistant
<Z_499><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?<|im_end|>
<|im_start|>assistant
<Z_439><Z_339><Z_495><Z_420><Z_383><Z_403><Z_415><Z_402><Z_414><Z_83><Z_424><ANSWER>

Question: A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.<|im_end|>
<|im_start|>assistant
<Z_312><ANSWER>

Question:
Define a function $f$ as follows: $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$,
\[f(n)=f(n-1)+f(n-2)+(2n).\]
Calculate $f(10)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|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
A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|im_end|>
<|im_start|>assistant
<Z_195><Z_100><Z_409><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|im_end|>
<|im_start|>assistant
<Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?<|im_end|>
<|im_start|>assistant
<Z_99><Z_218><Z_216><ANSWER>

Question: A movie that's 1.5 hours long is being replayed a certain number of times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. The movie theater operates for 11 hours each day. How many times is the movie replayed each day?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_176><Z_376><Z_332><Z_275><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_283><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$<|im_end|>
<|im_start|>assistant
<Z_119><Z_67><Z_271><Z_373><ANSWER>

Question: If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|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
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|im_end|>
<|im_start|>assistant
<Z_1><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?<|im_end|>
<|im_start|>assistant
<Z_425><ANSWER>

Question: A candy store has 6 boxes of chocolates. Each box has 500 pieces. How many pieces are there altogether in the boxes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_358><Z_325><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_151><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.<|im_end|>
<|im_start|>assistant
<Z_366><Z_150><ANSWER>

Question: Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 4 adjacent digits sum to at least 2. Find the number of elements in $\mathcal{T}_{13}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_195><Z_287><Z_488><Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?<|im_end|>
<|im_start|>assistant
<Z_468><Z_450><Z_252><Z_238><ANSWER>

Question: A train covers a distance at a certain speed in 3 hours. To cover the same distance in 1 hour, it must travel at a speed of 450 kmph. What is the initial speed of the train?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_66><Z_325><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_240><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?<|im_end|>
<|im_start|>assistant
<Z_36><Z_158><ANSWER>

Question: On graph paper, a $12 \times 12$ square is drawn. What is the minimum number of its cells that need to be shaded so that a figure of the type shown in Figure 10 cannot be placed in the unshaded area?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_5><Z_437><Z_113><Z_462><Z_157><Z_311><Z_295><Z_408><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?<|im_end|>
<|im_start|>assistant
<Z_266><Z_492><Z_403><Z_255><Z_204><Z_318><Z_270><Z_386><ANSWER>

Question: How many ways are there to arrange the letters of the word $\text{CO}_1\text{M}_1\text{P}_2\text{U}_2\text{T}_3\text{E}_3$, assuming each subscripted letter is distinct?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_195><Z_369><Z_191><Z_231><Z_57><Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?<|im_end|>
<|im_start|>assistant
<Z_353><Z_442><Z_46><Z_497><Z_99><Z_452><ANSWER>

Question: James has 18 chocolate bars to sell for the swim team. He sold some last week, 7 this week, and he needs to sell 6 more. How many chocolate bars did James sell last week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|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
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|im_end|>
<|im_start|>assistant
<Z_195><Z_409><Z_457><Z_153><Z_192><Z_293><Z_364><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|im_end|>
<|im_start|>assistant
<Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?<|im_end|>
<|im_start|>assistant
<Z_481><Z_162><Z_332><Z_92><Z_181><Z_95><Z_301><ANSWER>

Question: Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<Z_5><Z_366><Z_156><Z_302><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<Z_109><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?<|im_end|>
<|im_start|>assistant
<Z_511><Z_27><Z_459><Z_190><ANSWER>

Question: What is the coefficient of \(x^4\) in the expansion of \((x + 3\sqrt{2})^8\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|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
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|im_end|>
<|im_start|>assistant
<Z_358><Z_73><Z_116><Z_184><Z_509><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|im_end|>
<|im_start|>assistant
<Z_339><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?<|im_end|>
<|im_start|>assistant
<Z_91><Z_407><Z_267><Z_84><Z_434><ANSWER>

Question: To let Ruth know how long her fish will live, her father tells her that well-cared fish can live 2 years longer than dogs live. On average, dogs live 4 times as long as hamsters live. And hamsters live an average of 2.5 years. Her father keeps track of the life-spans of fish, dogs, and hamsters in months. How long can a fish live in months when well-cared for?