<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|im_end|>
<|im_start|>assistant
<Z_40><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student added seven to both the numerator and denominator of a fraction, changing the fraction's value to $\frac{1}{3}$. If the original numerator was a 3, what was the original denominator?<|im_end|>
<|im_start|>assistant
<Z_35><ANSWER>

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


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_29><Z_124><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_64><Z_111><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$?<|im_end|>
<|im_start|>assistant
<Z_124><Z_22><Z_84><ANSWER>

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


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_94><Z_63><Z_68><Z_107><Z_40><Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two family reunions are happening at the Leela Hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. 50 people attend the Oates reunion, some people attend the Hall reunion, and 12 people attend both reunions. How many people attend the Hall reunion?<|im_end|>
<|im_start|>assistant
<Z_67><Z_39><Z_114><Z_75><Z_81><Z_67><ANSWER>

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


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

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

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

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_94><Z_68><Z_110><Z_97><Z_33><Z_39><Z_87><Z_41><Z_125><Z_65><Z_91><Z_34><Z_39><Z_67><Z_96><Z_53><Z_90><Z_96><Z_54><Z_87><Z_96><ANSWER>

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

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_38><ANSWER>

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

*2016 CCA Math Bonanza Team #3*<|im_end|>
<|im_start|>assistant
<Z_22><Z_34><Z_85><Z_102><Z_58><Z_11><Z_90><Z_32><Z_66><Z_121><Z_98><Z_46><Z_25><Z_7><Z_51><Z_90><Z_107><Z_40><Z_119><Z_50><Z_70><ANSWER>

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

*2016 CCA Math Bonanza Team #3*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_94><Z_108><Z_110><Z_30><Z_62><Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many even integers are there between $\frac{23}{5}$ and $\frac{47}{3}$?<|im_end|>
<|im_start|>assistant
<Z_12><Z_112><Z_56><Z_9><Z_23><Z_37><ANSWER>

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


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_57><Z_1><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_10><Z_74><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
an outlet pipe empties a tank which is full in 5 hours . if the inlet pipe is kept open , which lets water in at the rate of 4 litres / min then outlet pipe would take 3 hours longer . find the capacity of the tank .<|im_end|>
<|im_start|>assistant
<Z_116><ANSWER>

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


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A hospital has 12 internal medicine doctors and 8 surgeons. Now, 5 doctors are to be selected to join the disaster relief medical team, where:

(1) A certain internal medicine doctor, A, and a certain surgeon, B, must participate. How many different selection methods are there?

(2) Both A and B cannot participate. How many selection methods are there?

(3) At least one of A and B must participate. How many selection methods are there?

(4) There must be at least one internal medicine doctor and one surgeon in the team. How many selection methods are there?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A hospital has 12 internal medicine doctors and 8 surgeons. Now, 5 doctors are to be selected to join the disaster relief medical team, where:

(1) A certain internal medicine doctor, A, and a certain surgeon, B, must participate. How many different selection methods are there?

(2) Both A and B cannot participate. How many selection methods are there?

(3) At least one of A and B must participate. How many selection methods are there?

(4) There must be at least one internal medicine doctor and one surgeon in the team. How many selection methods are there?<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A hospital has 12 internal medicine doctors and 8 surgeons. Now, 5 doctors are to be selected to join the disaster relief medical team, where:

(1) A certain internal medicine doctor, A, and a certain surgeon, B, must participate. How many different selection methods are there?

(2) Both A and B cannot participate. How many selection methods are there?

(3) At least one of A and B must participate. How many selection methods are there?

(4) There must be at least one internal medicine doctor and one surgeon in the team. How many selection methods are there?<|im_end|>
<|im_start|>assistant
<Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A hospital has 12 internal medicine doctors and 8 surgeons. Now, 5 doctors are to be selected to join the disaster relief medical team, where:

(1) A certain internal medicine doctor, A, and a certain surgeon, B, must participate. How many different selection methods are there?

(2) Both A and B cannot participate. How many selection methods are there?

(3) At least one of A and B must participate. How many selection methods are there?

(4) There must be at least one internal medicine doctor and one surgeon in the team. How many selection methods are there?<|im_end|>
<|im_start|>assistant
<Z_60><ANSWER>

Question:
A hospital has 12 internal medicine doctors and 8 surgeons. Now, 5 doctors are to be selected to join the disaster relief medical team, where:

(1) A certain internal medicine doctor, A, and a certain surgeon, B, must participate. How many different selection methods are there?

(2) Both A and B cannot participate. How many selection methods are there?

(3) At least one of A and B must participate. How many selection methods are there?

(4) There must be at least one internal medicine doctor and one surgeon in the team. How many selection methods are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car travels due east at 1 mile per minute on a long, straight road. Simultaneously, a circular storm, whose radius is 75 miles, moves northeast at $1 mile per minute$. At time $t=0$, the center of the storm is 130 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2)$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car travels due east at 1 mile per minute on a long, straight road. Simultaneously, a circular storm, whose radius is 75 miles, moves northeast at $1 mile per minute$. At time $t=0$, the center of the storm is 130 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2)$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car travels due east at 1 mile per minute on a long, straight road. Simultaneously, a circular storm, whose radius is 75 miles, moves northeast at $1 mile per minute$. At time $t=0$, the center of the storm is 130 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2)$.<|im_end|>
<|im_start|>assistant
<Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car travels due east at 1 mile per minute on a long, straight road. Simultaneously, a circular storm, whose radius is 75 miles, moves northeast at $1 mile per minute$. At time $t=0$, the center of the storm is 130 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2)$.<|im_end|>
<|im_start|>assistant
<Z_52><Z_29><ANSWER>

Question: A car travels due east at 1 mile per minute on a long, straight road. Simultaneously, a circular storm, whose radius is 75 miles, moves northeast at $1 mile per minute$. At time $t=0$, the center of the storm is 130 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

Question: Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $M$ be the number of positive integers that are less than or equal to $4095$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $M$ be the number of positive integers that are less than or equal to $4095$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_106><Z_41><Z_121><Z_63><Z_62><Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $M$ be the number of positive integers that are less than or equal to $4095$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $M$ be the number of positive integers that are less than or equal to $4095$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.<|im_end|>
<|im_start|>assistant
<Z_11><Z_110><Z_23><Z_22><Z_65><Z_44><Z_119><ANSWER>

Question: Let $M$ be the number of positive integers that are less than or equal to $4095$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The congruent sides of an isosceles triangle are each 7 cm long, and the perimeter is 25 cm. In centimeters, what is the length of the base?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The congruent sides of an isosceles triangle are each 7 cm long, and the perimeter is 25 cm. In centimeters, what is the length of the base?<|im_end|>
<|im_start|>assistant
<Z_57><Z_74><Z_82><Z_38><Z_38><Z_63><Z_62><Z_62><Z_34><Z_95><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The congruent sides of an isosceles triangle are each 7 cm long, and the perimeter is 25 cm. In centimeters, what is the length of the base?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The congruent sides of an isosceles triangle are each 7 cm long, and the perimeter is 25 cm. In centimeters, what is the length of the base?<|im_end|>
<|im_start|>assistant
<Z_23><Z_17><Z_6><Z_111><Z_46><Z_25><Z_30><Z_51><Z_40><Z_21><ANSWER>

Question: The congruent sides of an isosceles triangle are each 7 cm long, and the perimeter is 25 cm. In centimeters, what is the length of the base?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the least positive value of $t$ such that
\[\arcsin (\sin 3\alpha), \ \arcsin (\sin 6\alpha), \ \arccos (\cos 10\alpha), \ \arcsin (\sin t\alpha)\]
is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{3}.\]<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the least positive value of $t$ such that
\[\arcsin (\sin 3\alpha), \ \arcsin (\sin 6\alpha), \ \arccos (\cos 10\alpha), \ \arcsin (\sin t\alpha)\]
is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{3}.\]<|im_end|>
<|im_start|>assistant
<Z_29><Z_29><Z_97><Z_87><Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the least positive value of $t$ such that
\[\arcsin (\sin 3\alpha), \ \arcsin (\sin 6\alpha), \ \arccos (\cos 10\alpha), \ \arcsin (\sin t\alpha)\]
is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{3}.\]<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the least positive value of $t$ such that
\[\arcsin (\sin 3\alpha), \ \arcsin (\sin 6\alpha), \ \arccos (\cos 10\alpha), \ \arcsin (\sin t\alpha)\]
is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{3}.\]<|im_end|>
<|im_start|>assistant
<Z_21><Z_8><Z_68><Z_126><Z_91><ANSWER>

Question:
Compute the least positive value of $t$ such that
\[\arcsin (\sin 3\alpha), \ \arcsin (\sin 6\alpha), \ \arccos (\cos 10\alpha), \ \arcsin (\sin t\alpha)\]
is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{3}.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one seventh of $N$?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one seventh of $N$?<|im_end|>
<|im_start|>assistant
<Z_113><Z_79><Z_27><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one seventh of $N$?<|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
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one seventh of $N$?<|im_end|>
<|im_start|>assistant
<Z_37><Z_37><Z_51><Z_66><ANSWER>

Question: How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one seventh of $N$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving in opposite directions at 60 kmph and 90 kmph. Their lengths are 1.10 km and 0.9 km respectively. How long does it take for the slower train to cross the faster train in seconds?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving in opposite directions at 60 kmph and 90 kmph. Their lengths are 1.10 km and 0.9 km respectively. How long does it take for the slower train to cross the faster train in seconds?<|im_end|>
<|im_start|>assistant
<Z_94><Z_51><Z_72><Z_43><Z_41><Z_78><Z_125><Z_33><Z_124><Z_70><Z_117><Z_72><Z_63><Z_52><Z_78><Z_70><Z_75><Z_117><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two trains are moving in opposite directions at 60 kmph and 90 kmph. Their lengths are 1.10 km and 0.9 km respectively. How long does it take for the slower train to cross the faster train in seconds?<|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
Two trains are moving in opposite directions at 60 kmph and 90 kmph. Their lengths are 1.10 km and 0.9 km respectively. How long does it take for the slower train to cross the faster train in seconds?<|im_end|>
<|im_start|>assistant
<Z_110><Z_3><Z_90><Z_70><Z_42><Z_80><Z_46><Z_45><Z_29><Z_58><Z_20><Z_27><Z_100><Z_46><Z_44><Z_77><Z_9><Z_35><ANSWER>

Question: Two trains are moving in opposite directions at 60 kmph and 90 kmph. Their lengths are 1.10 km and 0.9 km respectively. How long does it take for the slower train to cross the faster train in seconds?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by the number of cells indicated in its cell (if the number is negative, the token moves to the left); it turns out that each cell again contains a token. This operation is repeated many times. For each token from the first segment, the number of operations is counted until it first returns to this segment. Prove that there are no more than 100 different numbers among the obtained counts.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by the number of cells indicated in its cell (if the number is negative, the token moves to the left); it turns out that each cell again contains a token. This operation is repeated many times. For each token from the first segment, the number of operations is counted until it first returns to this segment. Prove that there are no more than 100 different numbers among the obtained counts.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by the number of cells indicated in its cell (if the number is negative, the token moves to the left); it turns out that each cell again contains a token. This operation is repeated many times. For each token from the first segment, the number of operations is counted until it first returns to this segment. Prove that there are no more than 100 different numbers among the obtained counts.<|im_end|>
<|im_start|>assistant
<Z_62><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by the number of cells indicated in its cell (if the number is negative, the token moves to the left); it turns out that each cell again contains a token. This operation is repeated many times. For each token from the first segment, the number of operations is counted until it first returns to this segment. Prove that there are no more than 100 different numbers among the obtained counts.<|im_end|>
<|im_start|>assistant
<Z_55><ANSWER>

Question: A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by the number of cells indicated in its cell (if the number is negative, the token moves to the left); it turns out that each cell again contains a token. This operation is repeated many times. For each token from the first segment, the number of operations is counted until it first returns to this segment. Prove that there are no more than 100 different numbers among the obtained counts.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
mr yadav spends 60 % of his monthly salary on consumable items and 50 % of the remaining on clothes and transport . he saves the remaining amount . if his savings at the end of the year were 48456 , how much amount per month would he have spent on clothes and transport ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
mr yadav spends 60 % of his monthly salary on consumable items and 50 % of the remaining on clothes and transport . he saves the remaining amount . if his savings at the end of the year were 48456 , how much amount per month would he have spent on clothes and transport ?<|im_end|>
<|im_start|>assistant
<Z_57><Z_73><Z_100><Z_59><Z_47><Z_119><Z_117><Z_56><Z_16><Z_103><Z_124><Z_80><Z_75><Z_39><Z_10><Z_24><Z_5><Z_119><Z_71><Z_72><Z_99><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
mr yadav spends 60 % of his monthly salary on consumable items and 50 % of the remaining on clothes and transport . he saves the remaining amount . if his savings at the end of the year were 48456 , how much amount per month would he have spent on clothes and transport ?<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
mr yadav spends 60 % of his monthly salary on consumable items and 50 % of the remaining on clothes and transport . he saves the remaining amount . if his savings at the end of the year were 48456 , how much amount per month would he have spent on clothes and transport ?<|im_end|>
<|im_start|>assistant
<Z_77><Z_39><Z_9><Z_65><Z_66><Z_42><Z_115><Z_22><Z_11><Z_95><Z_106><Z_127><Z_123><Z_96><Z_127><Z_80><Z_3><Z_98><Z_30><Z_25><Z_76><Z_85><ANSWER>

Question: mr yadav spends 60 % of his monthly salary on consumable items and 50 % of the remaining on clothes and transport . he saves the remaining amount . if his savings at the end of the year were 48456 , how much amount per month would he have spent on clothes and transport ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum \(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}\) is represented as a fraction with the denominator \(45! = 1 \cdot 2 \cdots 45\). How many zeros (in decimal notation) does the numerator of this fraction end with?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum \(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}\) is represented as a fraction with the denominator \(45! = 1 \cdot 2 \cdots 45\). How many zeros (in decimal notation) does the numerator of this fraction end with?<|im_end|>
<|im_start|>assistant
<Z_94><Z_68><Z_41><Z_125><Z_107><Z_103><Z_120><Z_26><Z_99><Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum \(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}\) is represented as a fraction with the denominator \(45! = 1 \cdot 2 \cdots 45\). How many zeros (in decimal notation) does the numerator of this fraction end with?<|im_end|>
<|im_start|>assistant
<Z_13><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum \(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}\) is represented as a fraction with the denominator \(45! = 1 \cdot 2 \cdots 45\). How many zeros (in decimal notation) does the numerator of this fraction end with?<|im_end|>
<|im_start|>assistant
<Z_43><Z_6><Z_124><Z_91><Z_69><Z_50><Z_68><Z_118><Z_42><Z_115><ANSWER>

Question: The sum \(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}\) is represented as a fraction with the denominator \(45! = 1 \cdot 2 \cdots 45\). How many zeros (in decimal notation) does the numerator of this fraction end with?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some weeks in the summer, Erica treats herself to 1 ice cream cone from the ice cream truck. Monday, Wednesday and Friday she gets a $2.00 orange creamsicle. Tuesday and Thursday she gets a $1.50 ice cream sandwich. Saturday and Sunday she gets a $3.00 Nutty-Buddy. She spends $90 on ice cream. How many weeks does she buy ice cream?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some weeks in the summer, Erica treats herself to 1 ice cream cone from the ice cream truck. Monday, Wednesday and Friday she gets a $2.00 orange creamsicle. Tuesday and Thursday she gets a $1.50 ice cream sandwich. Saturday and Sunday she gets a $3.00 Nutty-Buddy. She spends $90 on ice cream. How many weeks does she buy ice cream?<|im_end|>
<|im_start|>assistant
<Z_113><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some weeks in the summer, Erica treats herself to 1 ice cream cone from the ice cream truck. Monday, Wednesday and Friday she gets a $2.00 orange creamsicle. Tuesday and Thursday she gets a $1.50 ice cream sandwich. Saturday and Sunday she gets a $3.00 Nutty-Buddy. She spends $90 on ice cream. How many weeks does she buy ice cream?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some weeks in the summer, Erica treats herself to 1 ice cream cone from the ice cream truck. Monday, Wednesday and Friday she gets a $2.00 orange creamsicle. Tuesday and Thursday she gets a $1.50 ice cream sandwich. Saturday and Sunday she gets a $3.00 Nutty-Buddy. She spends $90 on ice cream. How many weeks does she buy ice cream?<|im_end|>
<|im_start|>assistant
<Z_5><Z_30><ANSWER>

Question: For some weeks in the summer, Erica treats herself to 1 ice cream cone from the ice cream truck. Monday, Wednesday and Friday she gets a $2.00 orange creamsicle. Tuesday and Thursday she gets a $1.50 ice cream sandwich. Saturday and Sunday she gets a $3.00 Nutty-Buddy. She spends $90 on ice cream. How many weeks does she buy ice cream?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Worker A takes some hours to do a job. Worker B takes 10 hours to do the same job. Working together but independently, they can do the same job in 2.857142857142857 hours. How long does it take for Worker A to do the job alone?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Worker A takes some hours to do a job. Worker B takes 10 hours to do the same job. Working together but independently, they can do the same job in 2.857142857142857 hours. How long does it take for Worker A to do the job alone?<|im_end|>
<|im_start|>assistant
<Z_94><Z_1><Z_65><Z_114><Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Worker A takes some hours to do a job. Worker B takes 10 hours to do the same job. Working together but independently, they can do the same job in 2.857142857142857 hours. How long does it take for Worker A to do the job alone?<|im_end|>
<|im_start|>assistant
<Z_15><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Worker A takes some hours to do a job. Worker B takes 10 hours to do the same job. Working together but independently, they can do the same job in 2.857142857142857 hours. How long does it take for Worker A to do the job alone?<|im_end|>
<|im_start|>assistant
<Z_83><Z_96><Z_34><Z_92><Z_52><ANSWER>

Question: Worker A takes some hours to do a job. Worker B takes 10 hours to do the same job. Working together but independently, they can do the same job in 2.857142857142857 hours. How long does it take for Worker A to do the job alone?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $(4+2+6)\div3 - (2+1)\div3$ in simplest terms?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $(4+2+6)\div3 - (2+1)\div3$ in simplest terms?<|im_end|>
<|im_start|>assistant
<Z_57><Z_29><Z_38><Z_125><Z_65><Z_123><Z_64><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $(4+2+6)\div3 - (2+1)\div3$ in simplest terms?<|im_end|>
<|im_start|>assistant
<Z_25><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $(4+2+6)\div3 - (2+1)\div3$ in simplest terms?<|im_end|>
<|im_start|>assistant
<Z_65><Z_13><Z_36><Z_12><Z_77><Z_88><Z_46><Z_62><ANSWER>

Question: What is $(4+2+6)\div3 - (2+1)\div3$ in simplest terms?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ordered pairs $(a, b)$ that satisfy $a, b \in \{-1, 0, 1, 2\}$ and the quadratic equation $ax^2 + 2x + b = 0$ has a real solution.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ordered pairs $(a, b)$ that satisfy $a, b \in \{-1, 0, 1, 2\}$ and the quadratic equation $ax^2 + 2x + b = 0$ has a real solution.<|im_end|>
<|im_start|>assistant
<Z_104><Z_40><Z_65><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ordered pairs $(a, b)$ that satisfy $a, b \in \{-1, 0, 1, 2\}$ and the quadratic equation $ax^2 + 2x + b = 0$ has a real solution.<|im_end|>
<|im_start|>assistant
<Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of ordered pairs $(a, b)$ that satisfy $a, b \in \{-1, 0, 1, 2\}$ and the quadratic equation $ax^2 + 2x + b = 0$ has a real solution.<|im_end|>
<|im_start|>assistant
<Z_64><Z_75><Z_12><Z_82><ANSWER>

Question: Determine the number of ordered pairs $(a, b)$ that satisfy $a, b \in \{-1, 0, 1, 2\}$ and the quadratic equation $ax^2 + 2x + b = 0$ has a real solution.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of a group of 10 students is 14 years. If 5 more students join the group, the average age rises by 1 year. What is the average age of the new students?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of a group of 10 students is 14 years. If 5 more students join the group, the average age rises by 1 year. What is the average age of the new students?<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of a group of 10 students is 14 years. If 5 more students join the group, the average age rises by 1 year. What is the average age of the new students?<|im_end|>
<|im_start|>assistant
<Z_127><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average age of a group of 10 students is 14 years. If 5 more students join the group, the average age rises by 1 year. What is the average age of the new students?<|im_end|>
<|im_start|>assistant
<Z_5><Z_47><ANSWER>

Question: The average age of a group of 10 students is 14 years. If 5 more students join the group, the average age rises by 1 year. What is the average age of the new students?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of pens to pencils is 5 to 6. There are some more pencils than pens, and there are 54 pencils. How many more pencils are there than pens?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of pens to pencils is 5 to 6. There are some more pencils than pens, and there are 54 pencils. How many more pencils are there than pens?<|im_end|>
<|im_start|>assistant
<Z_94><Z_125><Z_82><Z_51><Z_126><Z_119><Z_125><Z_107><Z_126><Z_22><Z_98><Z_61><Z_22><Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of pens to pencils is 5 to 6. There are some more pencils than pens, and there are 54 pencils. How many more pencils are there than pens?<|im_end|>
<|im_start|>assistant
<Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio of pens to pencils is 5 to 6. There are some more pencils than pens, and there are 54 pencils. How many more pencils are there than pens?<|im_end|>
<|im_start|>assistant
<Z_70><Z_91><Z_47><Z_89><Z_21><Z_27><Z_20><Z_95><Z_6><Z_81><Z_126><Z_11><Z_0><Z_85><ANSWER>

Question: The ratio of pens to pencils is 5 to 6. There are some more pencils than pens, and there are 54 pencils. How many more pencils are there than pens?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
This year, a class of 56 students subscribed to the "Junior High School Mathematics Study" magazine. Among them, 25 boys and 15 girls subscribed to the magazine in the first half of the year, and 26 boys and 25 girls subscribed in the second half of the year. There were 23 boys who subscribed for the whole year. How many girls only subscribed to the magazine in the first half of the year?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
This year, a class of 56 students subscribed to the "Junior High School Mathematics Study" magazine. Among them, 25 boys and 15 girls subscribed to the magazine in the first half of the year, and 26 boys and 25 girls subscribed in the second half of the year. There were 23 boys who subscribed for the whole year. How many girls only subscribed to the magazine in the first half of the year?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
This year, a class of 56 students subscribed to the "Junior High School Mathematics Study" magazine. Among them, 25 boys and 15 girls subscribed to the magazine in the first half of the year, and 26 boys and 25 girls subscribed in the second half of the year. There were 23 boys who subscribed for the whole year. How many girls only subscribed to the magazine in the first half of the year?<|im_end|>
<|im_start|>assistant
<Z_6><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
This year, a class of 56 students subscribed to the "Junior High School Mathematics Study" magazine. Among them, 25 boys and 15 girls subscribed to the magazine in the first half of the year, and 26 boys and 25 girls subscribed in the second half of the year. There were 23 boys who subscribed for the whole year. How many girls only subscribed to the magazine in the first half of the year?<|im_end|>
<|im_start|>assistant
<Z_31><ANSWER>

Question: This year, a class of 56 students subscribed to the "Junior High School Mathematics Study" magazine. Among them, 25 boys and 15 girls subscribed to the magazine in the first half of the year, and 26 boys and 25 girls subscribed in the second half of the year. There were 23 boys who subscribed for the whole year. How many girls only subscribed to the magazine in the first half of the year?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The four positive integers $a,$ $b,$ $c,$ $d$ satisfy \[a \times b \times c \times d = 12!.\] Find the smallest possible value of $a + b + c + d.$<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The four positive integers $a,$ $b,$ $c,$ $d$ satisfy \[a \times b \times c \times d = 12!.\] Find the smallest possible value of $a + b + c + d.$<|im_end|>
<|im_start|>assistant
<Z_104><Z_86><Z_105><Z_80><Z_38><Z_107><Z_121><Z_41><Z_115><Z_59><Z_60><Z_60><Z_45><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The four positive integers $a,$ $b,$ $c,$ $d$ satisfy \[a \times b \times c \times d = 12!.\] Find the smallest possible value of $a + b + c + d.$<|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
The four positive integers $a,$ $b,$ $c,$ $d$ satisfy \[a \times b \times c \times d = 12!.\] Find the smallest possible value of $a + b + c + d.$<|im_end|>
<|im_start|>assistant
<Z_23><Z_106><Z_48><Z_6><Z_123><Z_1><Z_33><Z_29><Z_61><Z_79><Z_107><Z_103><Z_99><Z_57><ANSWER>

Question: The four positive integers $a,$ $b,$ $c,$ $d$ satisfy \[a \times b \times c \times d = 12!.\] Find the smallest possible value of $a + b + c + d.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thief is spotted by a policeman from a distance of 100 meters. When the policeman starts the chase, the thief also starts running. The speed of the policeman is 10 km/hr, and the thief will have run 400 meters before he is overtaken. What is the speed of the thief in km/hr?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thief is spotted by a policeman from a distance of 100 meters. When the policeman starts the chase, the thief also starts running. The speed of the policeman is 10 km/hr, and the thief will have run 400 meters before he is overtaken. What is the speed of the thief in km/hr?<|im_end|>
<|im_start|>assistant
<Z_104><Z_34><Z_81><Z_20><Z_72><Z_121><Z_58><Z_55><Z_75><Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thief is spotted by a policeman from a distance of 100 meters. When the policeman starts the chase, the thief also starts running. The speed of the policeman is 10 km/hr, and the thief will have run 400 meters before he is overtaken. What is the speed of the thief in km/hr?<|im_end|>
<|im_start|>assistant
<Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A thief is spotted by a policeman from a distance of 100 meters. When the policeman starts the chase, the thief also starts running. The speed of the policeman is 10 km/hr, and the thief will have run 400 meters before he is overtaken. What is the speed of the thief in km/hr?<|im_end|>
<|im_start|>assistant
<Z_109><Z_87><Z_26><Z_83><Z_21><Z_97><Z_55><Z_89><Z_37><Z_62><ANSWER>

Question: A thief is spotted by a policeman from a distance of 100 meters. When the policeman starts the chase, the thief also starts running. The speed of the policeman is 10 km/hr, and the thief will have run 400 meters before he is overtaken. What is the speed of the thief in km/hr?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A passenger train takes four hours less for a journey of 600 km if its speed is increased by 12 kmph from its normal speed. Determine the normal speed.<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A passenger train takes four hours less for a journey of 600 km if its speed is increased by 12 kmph from its normal speed. Determine the normal speed.<|im_end|>
<|im_start|>assistant
<Z_104><Z_78><Z_49><Z_38><Z_126><Z_49><Z_47><Z_70><Z_125><Z_90><Z_34><Z_93><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A passenger train takes four hours less for a journey of 600 km if its speed is increased by 12 kmph from its normal speed. Determine the normal speed.<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A passenger train takes four hours less for a journey of 600 km if its speed is increased by 12 kmph from its normal speed. Determine the normal speed.<|im_end|>
<|im_start|>assistant
<Z_11><Z_58><Z_61><Z_56><Z_26><Z_61><Z_42><Z_75><Z_18><Z_37><Z_63><Z_103><Z_92><ANSWER>

Question: A passenger train takes four hours less for a journey of 600 km if its speed is increased by 12 kmph from its normal speed. Determine the normal speed.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the average of first 12 even numbers is ?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the average of first 12 even numbers is ?<|im_end|>
<|im_start|>assistant
<Z_104><Z_108><Z_97><Z_41><Z_22><Z_99><Z_124><Z_34><Z_47><Z_90><Z_80><Z_55><Z_39><Z_48><Z_121><Z_70><Z_37><Z_120><Z_70><Z_93><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the average of first 12 even numbers is ?<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the average of first 12 even numbers is ?<|im_end|>
<|im_start|>assistant
<Z_8><Z_66><Z_75><Z_42><Z_90><Z_83><Z_99><Z_82><Z_12><Z_103><Z_73><Z_96><Z_76><Z_19><Z_111><Z_41><Z_21><Z_110><Z_87><Z_118><Z_104><ANSWER>

Question: the average of first 12 even numbers is ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequence \(\{a_n\}\), given that \(a_1 = 4\) and \(a_{n+1} = a_n + 2\) for \(n \in \mathbb{N}^*\):
(1) Find the general formula for the sequence \(\{a_n\}\).
(2) Let \(b_n = (\sqrt{2})^{a_n - 2} - 3n\), calculate \(\lvert b_1 \rvert + \lvert b_2 \rvert + \lvert b_3 \rvert + \ldots + \lvert b_{10} \rvert\).<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequence \(\{a_n\}\), given that \(a_1 = 4\) and \(a_{n+1} = a_n + 2\) for \(n \in \mathbb{N}^*\):
(1) Find the general formula for the sequence \(\{a_n\}\).
(2) Let \(b_n = (\sqrt{2})^{a_n - 2} - 3n\), calculate \(\lvert b_1 \rvert + \lvert b_2 \rvert + \lvert b_3 \rvert + \ldots + \lvert b_{10} \rvert\).<|im_end|>
<|im_start|>assistant
<Z_113><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequence \(\{a_n\}\), given that \(a_1 = 4\) and \(a_{n+1} = a_n + 2\) for \(n \in \mathbb{N}^*\):
(1) Find the general formula for the sequence \(\{a_n\}\).
(2) Let \(b_n = (\sqrt{2})^{a_n - 2} - 3n\), calculate \(\lvert b_1 \rvert + \lvert b_2 \rvert + \lvert b_3 \rvert + \ldots + \lvert b_{10} \rvert\).<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequence \(\{a_n\}\), given that \(a_1 = 4\) and \(a_{n+1} = a_n + 2\) for \(n \in \mathbb{N}^*\):
(1) Find the general formula for the sequence \(\{a_n\}\).
(2) Let \(b_n = (\sqrt{2})^{a_n - 2} - 3n\), calculate \(\lvert b_1 \rvert + \lvert b_2 \rvert + \lvert b_3 \rvert + \ldots + \lvert b_{10} \rvert\).<|im_end|>
<|im_start|>assistant
<Z_80><Z_53><ANSWER>

Question:
In the sequence \(\{a_n\}\), given that \(a_1 = 4\) and \(a_{n+1} = a_n + 2\) for \(n \in \mathbb{N}^*\):
(1) Find the general formula for the sequence \(\{a_n\}\).
(2) Let \(b_n = (\sqrt{2})^{a_n - 2} - 3n\), calculate \(\lvert b_1 \rvert + \lvert b_2 \rvert + \lvert b_3 \rvert + \ldots + \lvert b_{10} \rvert\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The h.c.f. of two numbers is 23 and the other two factors of their l.c.m. are some numbers. The larger of the two numbers is 322. If one of the other factors is 13, what is the other factor?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The h.c.f. of two numbers is 23 and the other two factors of their l.c.m. are some numbers. The larger of the two numbers is 322. If one of the other factors is 13, what is the other factor?<|im_end|>
<|im_start|>assistant
<Z_57><Z_97><Z_107><Z_43><Z_114><Z_10><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The h.c.f. of two numbers is 23 and the other two factors of their l.c.m. are some numbers. The larger of the two numbers is 322. If one of the other factors is 13, what is the other factor?<|im_end|>
<|im_start|>assistant
<Z_44><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The h.c.f. of two numbers is 23 and the other two factors of their l.c.m. are some numbers. The larger of the two numbers is 322. If one of the other factors is 13, what is the other factor?<|im_end|>
<|im_start|>assistant
<Z_89><Z_77><Z_34><Z_70><Z_109><Z_106><Z_91><ANSWER>

Question: The h.c.f. of two numbers is 23 and the other two factors of their l.c.m. are some numbers. The larger of the two numbers is 322. If one of the other factors is 13, what is the other factor?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
You start out with some pens. Mike gives you another 20 pens. Since her father makes pens, Cindy decides to double your pens. Since you're nice, you give Sharon 10 pens. How many pens did you have at the beginning before Mike gave you any pens?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
You start out with some pens. Mike gives you another 20 pens. Since her father makes pens, Cindy decides to double your pens. Since you're nice, you give Sharon 10 pens. How many pens did you have at the beginning before Mike gave you any pens?<|im_end|>
<|im_start|>assistant
<Z_106><Z_103><Z_80><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
You start out with some pens. Mike gives you another 20 pens. Since her father makes pens, Cindy decides to double your pens. Since you're nice, you give Sharon 10 pens. How many pens did you have at the beginning before Mike gave you any pens?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
You start out with some pens. Mike gives you another 20 pens. Since her father makes pens, Cindy decides to double your pens. Since you're nice, you give Sharon 10 pens. How many pens did you have at the beginning before Mike gave you any pens?<|im_end|>
<|im_start|>assistant
<Z_105><Z_36><Z_1><Z_30><ANSWER>

Question: You start out with some pens. Mike gives you another 20 pens. Since her father makes pens, Cindy decides to double your pens. Since you're nice, you give Sharon 10 pens. How many pens did you have at the beginning before Mike gave you any pens?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 0.5% of a = 70 paise, what is the value of a?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 0.5% of a = 70 paise, what is the value of a?<|im_end|>
<|im_start|>assistant
<Z_94><Z_13><Z_13><Z_87><Z_64><Z_64><Z_79><Z_117><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 0.5% of a = 70 paise, what is the value of a?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If 0.5% of a = 70 paise, what is the value of a?<|im_end|>
<|im_start|>assistant
<Z_86><Z_64><Z_33><Z_74><Z_73><Z_43><Z_20><Z_117><Z_73><ANSWER>

Question: If 0.5% of a = 70 paise, what is the value of a?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The set $T = \{1, 2, 3, \ldots, 59, 60\}$ contains the first 60 positive integers. After removing multiples of 4 and multiples of 5, how many integers remain in the set $T$?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The set $T = \{1, 2, 3, \ldots, 59, 60\}$ contains the first 60 positive integers. After removing multiples of 4 and multiples of 5, how many integers remain in the set $T$?<|im_end|>
<|im_start|>assistant
<Z_113><Z_70><Z_36><Z_62><Z_108><Z_70><Z_121><Z_91><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The set $T = \{1, 2, 3, \ldots, 59, 60\}$ contains the first 60 positive integers. After removing multiples of 4 and multiples of 5, how many integers remain in the set $T$?<|im_end|>
<|im_start|>assistant
<Z_7><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The set $T = \{1, 2, 3, \ldots, 59, 60\}$ contains the first 60 positive integers. After removing multiples of 4 and multiples of 5, how many integers remain in the set $T$?<|im_end|>
<|im_start|>assistant
<Z_3><Z_102><Z_97><Z_62><Z_78><Z_94><Z_55><Z_74><ANSWER>

Question: The set $T = \{1, 2, 3, \ldots, 59, 60\}$ contains the first 60 positive integers. After removing multiples of 4 and multiples of 5, how many integers remain in the set $T$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_57><Z_49><Z_87><Z_11><Z_82><Z_80><Z_59><Z_88><Z_92><Z_98><Z_126><Z_123><Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_31><Z_92><Z_59><Z_105><Z_27><Z_49><Z_93><Z_68><Z_26><Z_11><Z_79><Z_29><Z_13><ANSWER>

Question:
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the smallest positive integer \( n \geq 3 \) for which
\[ A \equiv 2^{10 n} \ (\bmod 2^{170}) \]
where \( A \) denotes the result when the numbers \( 2^{10}, 2^{20}, \ldots, 2^{10 n} \) are written in decimal notation and concatenated (for example, if \( n=2 \) we have \( A=10241048576 \)).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the smallest positive integer \( n \geq 3 \) for which
\[ A \equiv 2^{10 n} \ (\bmod 2^{170}) \]
where \( A \) denotes the result when the numbers \( 2^{10}, 2^{20}, \ldots, 2^{10 n} \) are written in decimal notation and concatenated (for example, if \( n=2 \) we have \( A=10241048576 \)).<|im_end|>
<|im_start|>assistant
<Z_94><Z_70><Z_30><Z_51><Z_119><Z_31><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the smallest positive integer \( n \geq 3 \) for which
\[ A \equiv 2^{10 n} \ (\bmod 2^{170}) \]
where \( A \) denotes the result when the numbers \( 2^{10}, 2^{20}, \ldots, 2^{10 n} \) are written in decimal notation and concatenated (for example, if \( n=2 \) we have \( A=10241048576 \)).<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the smallest positive integer \( n \geq 3 \) for which
\[ A \equiv 2^{10 n} \ (\bmod 2^{170}) \]
where \( A \) denotes the result when the numbers \( 2^{10}, 2^{20}, \ldots, 2^{10 n} \) are written in decimal notation and concatenated (for example, if \( n=2 \) we have \( A=10241048576 \)).<|im_end|>
<|im_start|>assistant
<Z_41><Z_127><Z_62><Z_101><Z_14><Z_10><ANSWER>

Question:
Determine the smallest positive integer \( n \geq 3 \) for which
\[ A \equiv 2^{10 n} \ (\bmod 2^{170}) \]
where \( A \) denotes the result when the numbers \( 2^{10}, 2^{20}, \ldots, 2^{10 n} \) are written in decimal notation and concatenated (for example, if \( n=2 \) we have \( A=10241048576 \)).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James catches 3 kinds of fish. He catches some pounds of trout, 50% more pounds of salmon, and twice as much Tuna. He caught 1100 pounds of fish. How many pounds of trout did he catch?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James catches 3 kinds of fish. He catches some pounds of trout, 50% more pounds of salmon, and twice as much Tuna. He caught 1100 pounds of fish. How many pounds of trout did he catch?<|im_end|>
<|im_start|>assistant
<Z_113><Z_73><Z_108><Z_86><Z_79><Z_124><Z_125><Z_107><Z_125><Z_99><Z_5><Z_123><Z_98><Z_107><Z_119><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James catches 3 kinds of fish. He catches some pounds of trout, 50% more pounds of salmon, and twice as much Tuna. He caught 1100 pounds of fish. How many pounds of trout did he catch?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
James catches 3 kinds of fish. He catches some pounds of trout, 50% more pounds of salmon, and twice as much Tuna. He caught 1100 pounds of fish. How many pounds of trout did he catch?<|im_end|>
<|im_start|>assistant
<Z_3><Z_107><Z_52><Z_18><Z_84><Z_103><Z_72><Z_38><Z_115><Z_8><Z_19><Z_50><Z_38><Z_50><Z_21><Z_8><ANSWER>

Question: James catches 3 kinds of fish. He catches some pounds of trout, 50% more pounds of salmon, and twice as much Tuna. He caught 1100 pounds of fish. How many pounds of trout did he catch?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the first day of her vacation, Louisa traveled 200 miles. On the second day, traveling at the same average speed, she traveled some miles. The 200-mile trip took 3 hours less than the second day's trip. Her average speed was 50 miles per hour. How many miles did Louisa travel on the second day?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the first day of her vacation, Louisa traveled 200 miles. On the second day, traveling at the same average speed, she traveled some miles. The 200-mile trip took 3 hours less than the second day's trip. Her average speed was 50 miles per hour. How many miles did Louisa travel on the second day?<|im_end|>
<|im_start|>assistant
<Z_57><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the first day of her vacation, Louisa traveled 200 miles. On the second day, traveling at the same average speed, she traveled some miles. The 200-mile trip took 3 hours less than the second day's trip. Her average speed was 50 miles per hour. How many miles did Louisa travel on the second day?<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the first day of her vacation, Louisa traveled 200 miles. On the second day, traveling at the same average speed, she traveled some miles. The 200-mile trip took 3 hours less than the second day's trip. Her average speed was 50 miles per hour. How many miles did Louisa travel on the second day?<|im_end|>
<|im_start|>assistant
<Z_97><Z_59><ANSWER>

Question: On the first day of her vacation, Louisa traveled 200 miles. On the second day, traveling at the same average speed, she traveled some miles. The 200-mile trip took 3 hours less than the second day's trip. Her average speed was 50 miles per hour. How many miles did Louisa travel on the second day?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe took 4 equally weighted tests and was allowed to drop his lowest score, which was 25. After dropping the lowest score, his average test score improved to 45. What was his average test score before dropping the lowest score?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe took 4 equally weighted tests and was allowed to drop his lowest score, which was 25. After dropping the lowest score, his average test score improved to 45. What was his average test score before dropping the lowest score?<|im_end|>
<|im_start|>assistant
<Z_27><Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe took 4 equally weighted tests and was allowed to drop his lowest score, which was 25. After dropping the lowest score, his average test score improved to 45. What was his average test score before dropping the lowest score?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joe took 4 equally weighted tests and was allowed to drop his lowest score, which was 25. After dropping the lowest score, his average test score improved to 45. What was his average test score before dropping the lowest score?<|im_end|>
<|im_start|>assistant
<Z_101><Z_127><ANSWER>

Question: Joe took 4 equally weighted tests and was allowed to drop his lowest score, which was 25. After dropping the lowest score, his average test score improved to 45. What was his average test score before dropping the lowest score?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sucrose solution contains 10 grams of sucrose per a certain volume of solution. If 60 cubic centimeters of the solution were poured into an empty container, there would be 6 grams of sucrose in the container. What is the volume of the solution that contains 10 grams of sucrose?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sucrose solution contains 10 grams of sucrose per a certain volume of solution. If 60 cubic centimeters of the solution were poured into an empty container, there would be 6 grams of sucrose in the container. What is the volume of the solution that contains 10 grams of sucrose?<|im_end|>
<|im_start|>assistant
<Z_104><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sucrose solution contains 10 grams of sucrose per a certain volume of solution. If 60 cubic centimeters of the solution were poured into an empty container, there would be 6 grams of sucrose in the container. What is the volume of the solution that contains 10 grams of sucrose?<|im_end|>
<|im_start|>assistant
<Z_8><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sucrose solution contains 10 grams of sucrose per a certain volume of solution. If 60 cubic centimeters of the solution were poured into an empty container, there would be 6 grams of sucrose in the container. What is the volume of the solution that contains 10 grams of sucrose?<|im_end|>
<|im_start|>assistant
<Z_3><Z_45><ANSWER>

Question: A sucrose solution contains 10 grams of sucrose per a certain volume of solution. If 60 cubic centimeters of the solution were poured into an empty container, there would be 6 grams of sucrose in the container. What is the volume of the solution that contains 10 grams of sucrose?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio between Rahul and Deepak's ages is 4:3. After 6 years, Rahul's age will be 42 years. What is Deepak's present age?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio between Rahul and Deepak's ages is 4:3. After 6 years, Rahul's age will be 42 years. What is Deepak's present age?<|im_end|>
<|im_start|>assistant
<Z_94><Z_97><Z_91><Z_124><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio between Rahul and Deepak's ages is 4:3. After 6 years, Rahul's age will be 42 years. What is Deepak's present age?<|im_end|>
<|im_start|>assistant
<Z_6><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The ratio between Rahul and Deepak's ages is 4:3. After 6 years, Rahul's age will be 42 years. What is Deepak's present age?<|im_end|>
<|im_start|>assistant
<Z_15><Z_121><Z_113><Z_77><Z_79><ANSWER>

Question: The ratio between Rahul and Deepak's ages is 4:3. After 6 years, Rahul's age will be 42 years. What is Deepak's present age?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least number of some digits which is divisible by 15, 25, 40 and 75 is 9600. How many digits does this number have?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least number of some digits which is divisible by 15, 25, 40 and 75 is 9600. How many digits does this number have?<|im_end|>
<|im_start|>assistant
<Z_94><Z_51><Z_105><Z_43><Z_87><Z_98><Z_87><Z_90><Z_39><Z_107><Z_79><Z_90><Z_34><Z_19><Z_54><Z_119><Z_32><Z_75><Z_47><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least number of some digits which is divisible by 15, 25, 40 and 75 is 9600. How many digits does this number have?<|im_end|>
<|im_start|>assistant
<Z_33><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The least number of some digits which is divisible by 15, 25, 40 and 75 is 9600. How many digits does this number have?<|im_end|>
<|im_start|>assistant
<Z_36><Z_92><Z_23><Z_60><Z_107><Z_120><Z_7><Z_42><Z_74><Z_126><Z_63><Z_73><Z_42><Z_79><Z_17><Z_5><Z_116><Z_91><Z_58><Z_124><ANSWER>

Question: The least number of some digits which is divisible by 15, 25, 40 and 75 is 9600. How many digits does this number have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a_n\}$ where $a_1=3$, $a_2=7$, and for $n\geq1$ and $n\in\mathbb{N}^*$, $a_{n+2}$ is equal to the units digit of $a_n\cdot a_{n+1}$. Find the value of the 2010th term of this sequence.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a_n\}$ where $a_1=3$, $a_2=7$, and for $n\geq1$ and $n\in\mathbb{N}^*$, $a_{n+2}$ is equal to the units digit of $a_n\cdot a_{n+1}$. Find the value of the 2010th term of this sequence.<|im_end|>
<|im_start|>assistant
<Z_104><Z_110><Z_73><Z_64><Z_12><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a_n\}$ where $a_1=3$, $a_2=7$, and for $n\geq1$ and $n\in\mathbb{N}^*$, $a_{n+2}$ is equal to the units digit of $a_n\cdot a_{n+1}$. Find the value of the 2010th term of this sequence.<|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
Given a sequence $\{a_n\}$ where $a_1=3$, $a_2=7$, and for $n\geq1$ and $n\in\mathbb{N}^*$, $a_{n+2}$ is equal to the units digit of $a_n\cdot a_{n+1}$. Find the value of the 2010th term of this sequence.<|im_end|>
<|im_start|>assistant
<Z_99><Z_48><Z_55><Z_46><Z_106><Z_28><ANSWER>

Question: Given a sequence $\{a_n\}$ where $a_1=3$, $a_2=7$, and for $n\geq1$ and $n\in\mathbb{N}^*$, $a_{n+2}$ is equal to the units digit of $a_n\cdot a_{n+1}$. Find the value of the 2010th term of this sequence.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest common divisor of $8!$ and $7!$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest common divisor of $8!$ and $7!$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest common divisor of $8!$ and $7!$.<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest common divisor of $8!$ and $7!$.<|im_end|>
<|im_start|>assistant
<Z_74><Z_21><ANSWER>

Question: Find the greatest common divisor of $8!$ and $7!$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the sides \(BC\) and \(AD\) of a convex quadrilateral \(ABCD\), the midpoints \(M\) and \(N\) are marked, respectively. Segments \(MN\) and \(AC\) intersect at point \(O\), and it is given that \(MO = ON\). It is known that the area of triangle \(ABC\) is 2017. Find the area of quadrilateral \(ABCD\).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the sides \(BC\) and \(AD\) of a convex quadrilateral \(ABCD\), the midpoints \(M\) and \(N\) are marked, respectively. Segments \(MN\) and \(AC\) intersect at point \(O\), and it is given that \(MO = ON\). It is known that the area of triangle \(ABC\) is 2017. Find the area of quadrilateral \(ABCD\).<|im_end|>
<|im_start|>assistant
<Z_104><Z_27><Z_89><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the sides \(BC\) and \(AD\) of a convex quadrilateral \(ABCD\), the midpoints \(M\) and \(N\) are marked, respectively. Segments \(MN\) and \(AC\) intersect at point \(O\), and it is given that \(MO = ON\). It is known that the area of triangle \(ABC\) is 2017. Find the area of quadrilateral \(ABCD\).<|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
On the sides \(BC\) and \(AD\) of a convex quadrilateral \(ABCD\), the midpoints \(M\) and \(N\) are marked, respectively. Segments \(MN\) and \(AC\) intersect at point \(O\), and it is given that \(MO = ON\). It is known that the area of triangle \(ABC\) is 2017. Find the area of quadrilateral \(ABCD\).<|im_end|>
<|im_start|>assistant
<Z_42><Z_31><Z_44><Z_119><ANSWER>

Question: On the sides \(BC\) and \(AD\) of a convex quadrilateral \(ABCD\), the midpoints \(M\) and \(N\) are marked, respectively. Segments \(MN\) and \(AC\) intersect at point \(O\), and it is given that \(MO = ON\). It is known that the area of triangle \(ABC\) is 2017. Find the area of quadrilateral \(ABCD\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Coach Tim has a team of 11 players, some of them didn't receive cautions, the rest received one yellow card each. The whole team would collect 3 red cards, knowing that each red card corresponds to 2 yellow cards. How many players didn't receive cautions?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Coach Tim has a team of 11 players, some of them didn't receive cautions, the rest received one yellow card each. The whole team would collect 3 red cards, knowing that each red card corresponds to 2 yellow cards. How many players didn't receive cautions?<|im_end|>
<|im_start|>assistant
<Z_57><Z_79><Z_99><Z_121><Z_114><Z_73><Z_91><Z_76><Z_9><Z_26><Z_80><Z_96><Z_37><Z_56><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Coach Tim has a team of 11 players, some of them didn't receive cautions, the rest received one yellow card each. The whole team would collect 3 red cards, knowing that each red card corresponds to 2 yellow cards. How many players didn't receive cautions?<|im_end|>
<|im_start|>assistant
<Z_67><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Coach Tim has a team of 11 players, some of them didn't receive cautions, the rest received one yellow card each. The whole team would collect 3 red cards, knowing that each red card corresponds to 2 yellow cards. How many players didn't receive cautions?<|im_end|>
<|im_start|>assistant
<Z_92><Z_89><Z_85><Z_58><Z_42><Z_106><Z_127><Z_14><Z_84><Z_96><Z_118><Z_42><Z_30><Z_110><Z_9><ANSWER>

Question: Coach Tim has a team of 11 players, some of them didn't receive cautions, the rest received one yellow card each. The whole team would collect 3 red cards, knowing that each red card corresponds to 2 yellow cards. How many players didn't receive cautions?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number increased by 20% gives 480. If this new number, 480, is then decreased by 15% and multiplied by the square of a variable x, the result is equal to 5x^3 + 24x - 50. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number increased by 20% gives 480. If this new number, 480, is then decreased by 15% and multiplied by the square of a variable x, the result is equal to 5x^3 + 24x - 50. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_94><Z_50><Z_124><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number increased by 20% gives 480. If this new number, 480, is then decreased by 15% and multiplied by the square of a variable x, the result is equal to 5x^3 + 24x - 50. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A number increased by 20% gives 480. If this new number, 480, is then decreased by 15% and multiplied by the square of a variable x, the result is equal to 5x^3 + 24x - 50. What is the original number?<|im_end|>
<|im_start|>assistant
<Z_67><Z_67><Z_58><Z_103><ANSWER>

Question: A number increased by 20% gives 480. If this new number, 480, is then decreased by 15% and multiplied by the square of a variable x, the result is equal to 5x^3 + 24x - 50. 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 the equation "爱国 × 创新 × 包容 + 厚德 = 北京精神", each Chinese character represents a distinct digit from 0 to 9. What is the value of 厚德 when the four-digit number 北京精神 is at its maximum?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the equation "爱国 × 创新 × 包容 + 厚德 = 北京精神", each Chinese character represents a distinct digit from 0 to 9. What is the value of 厚德 when the four-digit number 北京精神 is at its maximum?<|im_end|>
<|im_start|>assistant
<Z_94><Z_99><Z_62><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the equation "爱国 × 创新 × 包容 + 厚德 = 北京精神", each Chinese character represents a distinct digit from 0 to 9. What is the value of 厚德 when the four-digit number 北京精神 is at its maximum?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the equation "爱国 × 创新 × 包容 + 厚德 = 北京精神", each Chinese character represents a distinct digit from 0 to 9. What is the value of 厚德 when the four-digit number 北京精神 is at its maximum?<|im_end|>
<|im_start|>assistant
<Z_75><Z_56><Z_57><ANSWER>

Question: In the equation "爱国 × 创新 × 包容 + 厚德 = 北京精神", each Chinese character represents a distinct digit from 0 to 9. What is the value of 厚德 when the four-digit number 北京精神 is at its maximum?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill can buy pans, pins, and pons for $\$3$, $\$5$, and $\$10$ each, respectively. What is the largest number of pons he can purchase if he must buy at least one of each item and will spend exactly $\$100$?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill can buy pans, pins, and pons for $\$3$, $\$5$, and $\$10$ each, respectively. What is the largest number of pons he can purchase if he must buy at least one of each item and will spend exactly $\$100$?<|im_end|>
<|im_start|>assistant
<Z_104><Z_100><Z_115><Z_30><Z_91><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill can buy pans, pins, and pons for $\$3$, $\$5$, and $\$10$ each, respectively. What is the largest number of pons he can purchase if he must buy at least one of each item and will spend exactly $\$100$?<|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bill can buy pans, pins, and pons for $\$3$, $\$5$, and $\$10$ each, respectively. What is the largest number of pons he can purchase if he must buy at least one of each item and will spend exactly $\$100$?<|im_end|>
<|im_start|>assistant
<Z_64><Z_27><Z_39><Z_120><Z_56><Z_49><ANSWER>

Question: Bill can buy pans, pins, and pons for $\$3$, $\$5$, and $\$10$ each, respectively. What is the largest number of pons he can purchase if he must buy at least one of each item and will spend exactly $\$100$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a regular decagon. How many triangles can be formed whose vertices are the vertices of the decagon?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a regular decagon. How many triangles can be formed whose vertices are the vertices of the decagon?<|im_end|>
<|im_start|>assistant
<Z_104><Z_40><Z_40><Z_37><Z_72><Z_36><Z_95><Z_34><Z_28><Z_98><Z_98><Z_28><Z_56><Z_79><Z_115><Z_93><Z_49><Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a regular decagon. How many triangles can be formed whose vertices are the vertices of the decagon?<|im_end|>
<|im_start|>assistant
<Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a regular decagon. How many triangles can be formed whose vertices are the vertices of the decagon?<|im_end|>
<|im_start|>assistant
<Z_90><Z_91><Z_34><Z_126><Z_23><Z_75><Z_37><Z_58><Z_94><Z_1><Z_59><Z_74><Z_1><Z_56><Z_12><Z_81><Z_7><Z_27><ANSWER>

Question: Consider a regular decagon. How many triangles can be formed whose vertices are the vertices of the decagon?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular farm has to be fenced one long side, one short side, and the diagonal. The cost of fencing is some amount per meter. The area of the farm is 1200 m², and the short side is 30 m long. The job would cost Rs. 1320. What is the cost of fencing per meter?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular farm has to be fenced one long side, one short side, and the diagonal. The cost of fencing is some amount per meter. The area of the farm is 1200 m², and the short side is 30 m long. The job would cost Rs. 1320. What is the cost of fencing per meter?<|im_end|>
<|im_start|>assistant
<Z_57><Z_107><Z_103><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular farm has to be fenced one long side, one short side, and the diagonal. The cost of fencing is some amount per meter. The area of the farm is 1200 m², and the short side is 30 m long. The job would cost Rs. 1320. What is the cost of fencing per meter?<|im_end|>
<|im_start|>assistant
<Z_3><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular farm has to be fenced one long side, one short side, and the diagonal. The cost of fencing is some amount per meter. The area of the farm is 1200 m², and the short side is 30 m long. The job would cost Rs. 1320. What is the cost of fencing per meter?<|im_end|>
<|im_start|>assistant
<Z_13><Z_52><Z_52><Z_90><ANSWER>

Question: A rectangular farm has to be fenced one long side, one short side, and the diagonal. The cost of fencing is some amount per meter. The area of the farm is 1200 m², and the short side is 30 m long. The job would cost Rs. 1320. What is the cost of fencing per meter?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac{1}{5}$?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac{1}{5}$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_89><Z_86><Z_80><Z_99><Z_91><Z_100><Z_34><Z_38><Z_96><Z_53><Z_101><Z_98><Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac{1}{5}$?<|im_end|>
<|im_start|>assistant
<Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac{1}{5}$?<|im_end|>
<|im_start|>assistant
<Z_78><Z_89><Z_55><Z_95><Z_84><Z_67><Z_8><Z_85><Z_116><Z_41><Z_122><Z_84><Z_15><Z_12><ANSWER>

Question: How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac{1}{5}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A profit of some amount is divided between X and Y in the ratio of 1/2 : 1/3. The difference between their profit shares is Rs. 100. What is the total profit amount?<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A profit of some amount is divided between X and Y in the ratio of 1/2 : 1/3. The difference between their profit shares is Rs. 100. What is the total profit amount?<|im_end|>
<|im_start|>assistant
<Z_104><Z_110><Z_59><Z_114><Z_119><Z_99><Z_38><Z_65><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A profit of some amount is divided between X and Y in the ratio of 1/2 : 1/3. The difference between their profit shares is Rs. 100. What is the total profit amount?<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A profit of some amount is divided between X and Y in the ratio of 1/2 : 1/3. The difference between their profit shares is Rs. 100. What is the total profit amount?<|im_end|>
<|im_start|>assistant
<Z_4><Z_79><Z_57><Z_19><Z_79><Z_69><Z_33><Z_121><Z_31><ANSWER>

Question: A profit of some amount is divided between X and Y in the ratio of 1/2 : 1/3. The difference between their profit shares is Rs. 100. What is the total profit amount?