inclusionAI/Ring-lite-rl-data huggingface dataset:
Let the complex number \( z \) satisfy the conditions that both the real and imaginary parts of \( \frac{z}{10} \) and \( \frac{10}{\bar{z}} \) are positive numbers less than 1. What is the area of the trajectory of the point corresponding to \( z \) in the complex plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75 - \frac{25}{2}\pi}
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
7.8. The distances from the midpoint of the height of a regular triangular pyramid to the lateral face and to the lateral edge are 2 and $\sqrt{14}$, respectively. Find the volume of the pyramid. If necessary, round your answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{533.38}
Tiffany attends the carnival and her mother gives her $3 to play on a ring toss game. For every red bucket she tosses a ring into she gets 2 points. For every green bucket she gets three points, and for every rare blue bucket, she gets 5 points. She gets zero points for a miss. Blue buckets are harder to hit, with only a 10% success rate. Every play costs her $1 and she gets 5 rings per play. She's played two games and already gotten 4 red buckets, 5 green buckets, and 1 blue bucket. If she plays until her money runs out, what is the most total points she can get for all three games?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
The owner of an apartment insured it for 3,750,000 rubles (the actual cost of the apartment is 7,500,000 rubles). The actual damage amounted to 2,750,000 rubles. The insurance compensation under the proportional liability system was 1,350,000 rubles. Determine the amount of the deductible provided for in the contract.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50000}
Given $p$: $x^{2}+2x-3 > 0$, $q$: $x > a$, and the sufficient but not necessary condition for $\lnot q$ is $\lnot p$, determine the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq 1}
设$S = \{ 1 , 2 , 3 , 4 \} , n$项数列$a _ { 1 } , a _ { 2 } , \cdots , a _ { n }$有以下性质：对于S的任何一个非空子集B，在该数列中有相邻的｜B｜项恰好组成集合B.求n的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
正整数$[ \frac { 1 0 ^ { 9 3 } } { 1 0 ^ { 3 1 } + 3 } ]$的末两位数字是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{08}
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
12. (10 points) In a math competition, each team can only score 0 points, 3 points, or 5 points per question. At the end of the competition, the total score of three teams is 32 points. If any team's total score can reach 32 points, how many different combinations of total scores are there for these three teams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
There are 12 numbers from a certain starting number to 47 that are exactly divisible by 3. What is the starting number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231361}
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
It is known that a cone with an acute triangular cross-section has a slant height of $4$. If the range of the areas of all cross-sections passing through the vertex of the cone is $(0,4\sqrt{3}]$, then the central angle of the sector in the lateral surface development of the cone is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
One end of the diameter of a semicircle coincides with the vertex of the angle at the base of an isosceles triangle, and the other end belongs to this base. Find the radius of the semicircle, given that it touches one of the legs of the triangle and divides the other leg into segments of 5 cm and 4 cm, measured from the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{\sqrt{11}} \text{ cm}}
Let $V$ be a 10 -dimensional real vector space and $U_{1}$ and $U_{2}$ two linear subspaces such that $U_{1} \subseteq U_{2}, \operatorname{dim}_{\mathbb{R}} U_{1}=3$ and $\operatorname{dim}_{\mathbb{R}} U_{2}=6$. Let $\mathcal{E}$ be the set of all linear maps $T: V \longrightarrow V$ which have $U_{1}$ and $U_{2}$ as invariant subspaces (i.e., $T\left(U_{1}\right) \subseteq U_{1}$ and $T\left(U_{2}\right) \subseteq U_{2}$ ). Calculate the dimension of $\mathcal{E}$ as a real vector space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given circle $O$, point $C$ is on one side of diameter $\overline{AB}$, and point $D$ is on the opposite side. If $\angle AOC = 40^{\circ}$, and $\angle DOB = 30^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{36}}
For positive integers $n$ and $k$, let $\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167}
Given the sets \( M = \{x, y, \lg(xy)\} \) and \( N = \{0, |x|, y\} \) with \( M = N \), find the value of \(\left(x + \frac{1}{y}\right) + \left(x^{2} + \frac{1}{y^{2}}\right) + \left(x^{3} + \frac{1}{y^{3}}\right) + \cdots + \left(x^{2001} + \frac{1}{y^{2001}}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
While playing a trivia game Adam scored 283 points total. He scored a certain amount of points each round and played about 4 rounds. How many points did Adam score each round?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
In triangle \(ABC\), \(AB = 6\), \(BC = 7\), and \(CA = 8\). Let \(D\), \(E\), and \(F\) be the midpoints of sides \(BC\), \(AC\), and \(AB\), respectively. Also, let \(O_A\), \(O_B\), and \(O_C\) be the circumcenters of triangles \(AFD\), \(BDE\), and \(CEF\), respectively. Find the area of triangle \(O_A O_B O_C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21 \sqrt{15}}{16}}
Teacher Xiaoyu stands in a 9-row by 9-column square formation, and she notices that there are 2 people directly in front of her. After the entire formation turns right, Teacher Xiaoyu notices that there are now 4 people directly in front of her. If the entire formation turns right again, how many people will Teacher Xiaoyu notice directly in front of her?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $n = (10^{2020} + 2020)^2$. Find the sum of all the digits of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
On a side of a triangle, four points \( K, P, H, \) and \( M \) are taken, which are respectively the midpoint of this side, the foot of the angle bisector of the opposite angle of the triangle, the point of tangency with this side of the inscribed circle in the triangle, and the foot of the corresponding altitude. Find \( KH \) if \( KP = a \) and \( KM = b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{ab}}
Seryozha chose two different natural numbers \(a\) and \(b\). He wrote down four numbers in his notebook: \(a, a+2, b\), and \(b+2\). Then he listed on the board all six pairwise products of the numbers from the notebook. What is the maximum number of perfect squares that can be among the numbers on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from around the United States to participate. Around lunch time, the politicians decide to play a game. In this game, players can score $19$ points for pegging the coordinator of the gathering with a spitball, $9$ points for downing an entire cup of the forum’s interpretation of coffee, or $8$ points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch.

What is the product of the two greatest scores that a player cannot score in this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1209}
Compute
\[
\left( 1 - \sin \frac {\pi}{8} \right) \left( 1 - \sin \frac {3\pi}{8} \right) \left( 1 - \sin \frac {5\pi}{8} \right) \left( 1 - \sin \frac {7\pi}{8} \right).
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
A polynomial $p(x) = \sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \le j \le n-1,$ and $a_{j+1}-a_j=-k$ for $n \le j \le 2n-2;$ we call $k$ the [i]step size[/i] of $p(x).$ A real number $k$ is called [i]good[/i] if there exists a mountainous polynomial $p(x)$ with step size $k$ such that $p(-3)=0.$ Let $S$ be the sum of all good numbers $k$ satisfying $k \ge 5$ or $k \le 3.$ If $S=\tfrac{b}{c}$ for relatively prime positive integers $b,c,$ find $b+c.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Two boys started running simultaneously around a circular track from the same point at speeds of 60 km/hr and 100 km/hr. They met for the first time anywhere on the track after 107.99136069114472 minutes, while moving in opposite directions. What is the length of the track?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{287.98}
Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{2} + \sqrt{3}}{6}}
Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number on the card. If the sum of these scores equals $2004$, the subset is called a "good" hand. How many "good" hands are there?

(2004 National Girls' Olympiad problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006009}
Find the extremum of the function $f(x) = (1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!})e^{-x}$ where $n \geq 4$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are 5 telephone poles, 2 of which are electrified and will cause birds to be electrocuted and fall if touched. Given that 5 birds randomly choose one of the telephone poles to rest on, what is the probability that only 2 of the telephone poles have birds on them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{342}{625}}
There is a greatest number which on dividing a certain number and 2037 leaves remainders 7 and 5 respectively. The greatest number is 2. What is the first number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There are three types of people, A, B, and C, with a total of 25 people. Type A people always tell the truth, Type B people always lie, and Type C people alternate between telling the truth and lying (for example, if a Type C person tells the truth this time, their next statement will certainly be a lie, and the statement after that will be true again).

The priest asks each person, "Are you a Type A person?" 17 people answer "yes."
The priest then asks each person, "Are you a Type C person?" 12 people answer "yes."
The priest finally asks each person, "Are you a Type B person?" 8 people answer "yes."

How many of the 25 people are Type C people?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A train of a certain length is running at 60 kmph. It takes 23.998080153587715 seconds to pass a platform 260 m long. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{139.97}
计算：125(7/8)/(11(3/4)-4(3/20)+2.25-(7/20))．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.25}
Given the numbers: one and nine zeros. It is allowed to select two numbers and replace each with their arithmetic mean. What is the smallest number that can appear in place of the one after a series of such operations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{512}}
Given the median AD and BE of $\triangle ABC$ intersect at K, and $AB = \sqrt{3}$, with points K, D, C, E being concyclic, find the length of CK.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$. What is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the magnitude of the star Altair is $0.75$ and the magnitude of the star Vega is $0$, determine the ratio of the luminosity of Altair to Vega.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{-\frac{3}{10}}}
Let $T$ be the triangle in the coordinate plane with vertices $(0,0)$, $(4,0)$, and $(0,3)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find how many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The degree-$6$ polynomial $f$ satisfies the following conditions:  
- $f(7) - f(1) = 1$,  
- $f(8) - f(2) = 16$,  
- $f(9) - f(3) = 81$,  
- $f(10) - f(4) = 256$,  
- $f(11) - f(5) = 625$.  

Compute $f(15) - f(-3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6723}
Given the universal set U = {x | x is a positive integer less than 9}, set M = {1, 2, 3}, and set N = {3, 4, 5, 6}, find the intersection of the complement of M in U and N. Express your answer as a set of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{4, 5, 6}}
Given that the "optimized sum" of the sequence $a_1, a_2,..., a_{2018}$ is 2019, calculate the "optimized sum" of the sequence $2, a_1, a_2,..., a_{2018}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{91}{5}}
In January, Luke had 48 dollars. He spent 30% of his money during February. In March, he spent another 11 dollars and received 21 dollars from his mom. If he saved 10% of his remaining money each month from April to June, how much money would he have at the end of June?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$31.79}
A triangular pyramid stack of apples is formed with a base consisting of 6 apples on each side. Each higher layer is formed by 3 apples per row, with the top layer consisting of a single apple. Calculate the total number of apples used in the entire stack.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Considering all numbers of the form $n = \left\lfloor \frac{k^3}{2012} \right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1995}
Sides \(\overline{AB}\) and \(\overline{AC}\) of equilateral triangle \(ABC\) are tangent to a circle at points \(B\) and \(C\) respectively. Find the fraction of the area of \(\triangle ABC\) that lies outside the circle. The original answer is in the form \(\frac{k}{m}-\frac{n\sqrt{3}\pi}{p}\). Please give the value of \(k + m + n + p\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
A house worth $9000 is sold by Mr. A to Mr. B at a 10% loss. Mr. B then sells the house back to Mr. A at a 10% gain. Calculate Mr. A's net loss or gain after both transactions. Express your answer as a dollar value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810}
Alice buys three burgers, two sodas, and one fry for \$4.20, and Bill buys two burgers, one soda, and two fries for \$3.40. Determine the cost of one fry if a burger costs 80 cents and a soda costs 60 cents.
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Answer: \boxed{60}
Let \( R \) be the region in the first quadrant bounded by the x-axis, the line \( 2y = x \), and the ellipse \( \frac{x^2}{9} + y^2 = 1 \). Let \( R' \) be the region in the first quadrant bounded by the y-axis, the line \( y = mx \), and the ellipse. Find \( m \) such that \( R \) and \( R' \) have the same area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{9}}
Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The numbers \( x, y, \) and \( z \) are such that \( \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1 \). What values can the expression \( \frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \) take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[
(x^2+xy+y^2)(y^2+yz+z^2) + (y^2+yz+z^2)(z^2+zx+x^2) + (z^2+zx+x^2)(x^2+xy+y^2) = 625+ \frac{3}{4}(xy+yz+zx)^2.
\]
Compute $|xy+yz+zx|$.
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Answer: \boxed{14}
For a Poisson distribution with parameter $\lambda$, calculate the following limit as $\lambda \rightarrow \infty$ and $k \rightarrow \infty$ such that $k/\lambda \rightarrow a$, where $a < 1$:

$$\lim \frac{\mathbb{P}(N < k)}{\mathbb{P}(N = k)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{1 - a}\)}
设 $a_{n}=\left(\frac{1+q}{1-q}\right) \cdot\left(\frac{1+q^{2}}{1-q^{2}}\right)^{\frac{1}{2}} \cdot\left(\frac{1+q^{4}}{1-q^{4}}\right)^{\frac{1}{4}} \cdots\left(\frac{1+q^{2^{n}}}{1-q^{2^{n}}}\right)^{\frac{1}{2^{n}}}$, 其中 $q=\frac{11}{13}$ 。记 $a=\lim _{n \rightarrow+\infty} a_{n}$ ，则 $[a]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
The initial terms of the sequence $\{a_n\}$ are 1, 3, 5, 7, 9, 11, 13,... In the sequence $\{b_n\}$, $b_1=a_1$, $b_2=a_2$, $b_8=a_8$,... What is the value of $b_{20}$?
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Answer: \boxed{39}
The sequence $\left\{a_{n}\right\}$ is: $1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,\cdots$. The sequence is formed as follows: start with $a_{1}=1$ and copy this term to obtain $a_{2}=1$. Then add the number 2 as $a_{3}=2$. Next, copy all terms $1,1,2$ and place them as $a_{4}, a_{5}, a_{6}$, then add the number 3 as $a_{7}=3$, and so on. What is $a_{2021}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An elevator starts from rest at the top of a building and travels to the bottom. The elevator accelerates from rest between 2 s and 4 s, then travels at a constant speed, and finally decelerates to a stop between 22 s and 24 s. At what time interval does the elevator have its maximum downward velocity? Express your answer as a time interval in seconds (s).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[4, 22]}
The dividend is 12, the divisor is 17, the quotient is some number, and the remainder is 8. What is the quotient?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}