inclusionAI/Ring-lite-rl-data huggingface dataset:
3. In triangle $A B C$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn, $L$ is the intersection point of segments $B_{1} C_{1}$ and $A A_{1}$, $K$ is the intersection point of segments $B_{1} A_{1}$ and $C C_{1}$. Find the ratio $L M: M K$, if $M$ is the intersection point of the angle bisector $B B_{1}$ with segment $L K$, and $A B: B C: A C=2: 3: 4$. (16 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{12}}
Given a positive integer \(n\), find the number of positive integer solutions \((x, y)\) to the equation
$$
\frac{1}{n}=\frac{1}{x}+\frac{1}{y} \quad(x \neq y)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d(n^2) - 1}
find the least number must be subtracted from 5474827 so that remaining no . is divisible by 12 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Over all real numbers \( x \) and \( y \) such that
\[ x^3 = 3x + y \quad \text{and} \quad y^3 = 3y + x \]
compute the sum of all possible values of \( x^2 + y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Pablo has 27 solid $1 \times 1 \times 1$ cubes that he assembles in a larger $3 \times 3 \times 3$ cube. If 10 of the smaller cubes are red, 9 are blue, and 8 are yellow, what is the smallest possible surface area of the larger cube that is red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
If $f(x)=\frac{1}{(1-x^{2011})^{\frac{1}{2011}}}$, then what is the number of solutions of the equation $\underbrace{f\circ f\circ\dots\circ f}_{2013} (x)=\{-x\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Helen the hippo and her friends are preparing for thanksgiving at Helen's house. Helen baked 519 chocolate chip cookies and 300 raisin cookies yesterday. And she baked 280 raisin cookies and some chocolate chip cookies this morning. Helen baked 20 more raisin cookies yesterday compared to today. How many chocolate chip cookies did Helen bake today?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{539}
Find the maximum possible sum of the lengths of disjoint intervals in $(0,1)$ such that no two points in the union are at a distance of $\frac{1}{6}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}$}
Let  $n$  be a positive integer, and let  $S_n = \{1, 2, \ldots, n\}$ . For a permutation  $\sigma$  of  $S_n$  and an integer  $a \in S_n$ , let  $d(a)$  be the least positive integer  $d$  for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or  $-1$  if no such integer exists). Compute the value of  $n$  for which there exists a permutation  $\sigma$  of  $S_n$  satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017,  \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\]

*Proposed by Michael Tang*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
On the health and wellness day, fruits were distributed among the students in a school so that each student would get two bananas, one apple, and an orange. On the particular day, 420 students were absent, which resulted in the remaining students receiving four extra bananas, two more apples, and an additional orange each. Considering that bananas were distributed first, followed by apples and then oranges, find the actual number of students in the school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
A steamboat travels from point $A$ to point $B$ in $a$ hours and back in $b$ hours. How many hours will a raft take to drift from $A$ to $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2ab}{b-a}}
求最大正整数 $m$, 使得可以在 $m$ 行 8列的方格表的每个方格中填人 C、G、M、O 这四个字母之一，并且具有如下性质：对于方格表的任意不同两行，至多存在一列，使得这两行在此列处的字母相同。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A survey of $120$ teachers determined the following:

$70$ had high blood pressure

$40$ had heart trouble

$20$ had both high blood pressure and heart trouble

What percent of the teachers surveyed had neither high blood pressure nor heart trouble?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Parker sleeps for 7 hours everyday. If today he woke up at 9 in the morning, by what time would he would go to bed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:00}
What are the last two digits of the decimal representation of \(9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let $a$, $b$, and $c$ be integers simultaneously satisfying the equations:
\[
4abc + a + b + c = 2018
\]
and
\[
ab + bc + ca = -507
\]
Find \(|a| + |b| + |c|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
已知[x]表示不超过x的最大整数，则方程$3 ^ { 2 x } - [ 1 0 \cdot 3 ^ { x + 1 } ] +$ $\sqrt { 3 ^ { 2 x } - [ 1 0 \cdot 3 ^ { x + 1 } ] + 8 2 } = - 8 0$的整数解的个数为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{455}
There are \(2n\) complex numbers satisfying \(z^{28} - z^8 - 1 = 0\). They have the form \(z_{m} = \cos \theta_{m} + i \sin \theta_{m}\), \(m = 1, 2, \cdots, 2n\), where \(0 \leqslant \theta_{1} < \theta_{2} < \cdots < \theta_{2n} < 2\pi\). Find the value of \(\theta_{1} + \theta_{2} + \cdots + \theta_{2n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14\pi}
Consider a counting problem involving an \(n \times m\) grid, where each square contains the number of rectangles that can be formed inside the grid with that square as part of its boundary. For instance, in a \(2 \times 3\) grid, a green square is contained in 8 rectangles. Let \(f(n, m)\) be the sum of these numbers for all squares in the grid. Find a general formula for \(f(n, m)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n+2}{3} \cdot \binom{m+2}{3}}
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
已知点A(1,1),$B ( \frac { 1 } { 2 } ,$ $, 0 ) , C ( \frac { 3 } { 2 } , 0 )$经过点A、B的直线和经过点A、C的直线与直线y=a(0&lt;a&lt;1)所围成的平面区域为G.已知平面矩形区域{(x,y)|0&lt;x&lt;2,0&lt;y&lt;1}中的任意一点进入区域G的可能性为$\frac { 1 } { 1 6 }$，则a=
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 1 } { 2 }$ }
6 阶置换群 S_6 中的 6 阶元素的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let $Q(x)$ be a polynomial satisfying the functional equation $Q(x)=Q\left(\frac{1}{1-x}\right)$ for all $x\neq0,1$. Suppose that $P(x)$ is a polynomial such that $P(x)=x(1-x)Q(x)$ for all real $x$. If $P(x)$ is a cubic polynomial, what is the degree of $Q(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given two positive numbers $a$, $b$ such that $a<b$. Let $A.M.$ be their arithmetic mean and let $G.M.$ be their positive geometric mean. Then $A.M.$ minus $G.M.$ is always less than:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(b-a)^2}{8a}}
Triangle $ABC$ has an area 1.  Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7 - 3 \sqrt{5}}{4}}
A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 5 greater than the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
OpenAI runs a robotics competition that limits the weight of each robot. Each robot can be no more than twice the minimum weight and no less than 5 pounds heavier than the standard robot. The standard robot weighs 100 pounds. Additionally, each robot must use a specific battery that weighs 20 pounds, and they can only use one battery. The robots must also carry a payload of 10 up to 25 pounds as part of the competition. The minimum weight of a robot is affected by the payload requirements. Taking into account the battery, payload, and aforementioned constraints, what is the maximum weight of a robot in the competition, including the battery and payload?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
For each integer $k \ge 2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. For example, $X_2 = 10241048576$. If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n \ge 2$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
There are 13 3-year-olds, 20 4-year-olds, 15 5-year-olds, 22 6-year-olds, 18 7-year-olds, 25 8-year-olds, 30 9-year-olds, and 16 10-year-olds at a particular Sunday school. Classes are organized as follows: 

Class 1: 3 and 4-year-olds
Class 2: 4 and 5-year-olds
Class 3: 5, 6, and 7-year-olds
Class 4: 8-year-olds
Class 5: 9 and 10-year-olds

Given this distribution of students, what is the average class size?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.8}
If x and y are positive integers, and y = x^2 + 8, what is the greatest common divisor that x and y cannot have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Rohan spends 30% of his salary on food, 20% on house rent, and the rest on other expenses. His entertainment expenses are half of his food expenses, while his conveyance expenses are 25% more than his entertainment expenses. He spends 5% of his salary on education and 10% of his salary on utilities. If his savings at the end of a month are Rs. 2500, then what is his monthly salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200,000}
In the village, three types of cultivated land are classified: Category X, Category Y, and Category Z. Each attracts farm tax rates of 45%, 55%, and 65%, respectively. The tax department succeeded in collecting a total of $6500 through the farm tax. Mr. Willam, a farmer in the village, owns lands of all three categories. He paid $1200 as farm tax for his Category X land, $1600 for his Category Y land, and $1900 for his Category Z land. 

Given the total farm tax from the village and Mr. Willam's farm tax payment for each category of land, find the percentage of total land Mr. Willam owns over the total taxable land in the village. Assume that the total taxable land in the village is 1000 acres.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72.31\%}
Find the number of roots of the equation

$$
Q_{7}(z) \equiv z^{7}-2z-5=0
$$

in the right half-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
March 9, 2014 is a Sunday. Based on this information, determine which day of the week occurs most frequently in the year 2014.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Wednesday}
## Task Condition

Find the derivative.

$$
y=\frac{1}{4 \sqrt{5}} \ln \frac{2+\sqrt{5} \tanh x}{2-\sqrt{5} \cdot \tanh x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4-\sinh^{2}x}}
作出正四面体每个面（三角形）的中位线，共得12条线段，在这些线段中，相互成异面直线的“线段对”有几个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Consider a rectangular grid of  $ 10 \times 10$  unit squares. We call a *ship* a figure made up of unit squares connected by common edges. We call a *fleet* a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given that \( x, y, z \in \mathbb{R}_{+} \), and \( s = \sqrt{x+2} + \sqrt{y+5} + \sqrt{z+10} \), \( t = \sqrt{x+1} + \sqrt{y+1} + \sqrt{z+1} \), find the minimum value of \( s^{2} - t^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Andrey found the product of all the numbers from 1 to 11 inclusive and wrote the result on the board. During a break, someone accidentally erased three digits, leaving the number $399 * 68 * *$. Help restore the missing digits without recalculating the product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39916800}
Rhonda invested $4725 for one year, part at a certain annual interest rate, and the balance at 8% annual interest. She earned twice as much interest from the higher interest rate investment as from the 8% investment. She invested $1925 at the higher interest rate and the rest at 8%. What was the higher annual interest rate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.27\%}
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $3:1$. Calculate the percent of the rectangle's area that is contained within the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.33\%}
A piece of rope is randomly cut into two parts. Find the probability that the longer segment is at least x times the shorter segment. Express your answer as a fraction or ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{x+1}}
Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159}
On the radius \( AO \) of a circle with center \( O \), point \( M \) is chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \( BC \), given that the radius of the circle is 15 and \(\sin \alpha = \frac{\sqrt{21}}{5}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the following limit:
$$\lim_{n \to \infty} \left[\prod_{k=1}^{n}\left(1+\frac{(k+1)^{n}}{n^{n+1}}\right)\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given $(1+x) + (1+x)^2 + \ldots + (1+x)^5 = a_0 + a_1(1-x) + a_2(1-x)^2 + \ldots + a_5(1-x)^5$, calculate the value of $a_1 + a_2 + a_3 + a_4 + a_5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-57}
A math test consists of 12 multiple-choice questions, each worth 5 points. It is known that a student is confident in correctly answering 6 of these questions. For another three questions, the student can eliminate one incorrect option. For two questions, the student can eliminate two incorrect options. For the last question, due to a lack of understanding, the student has to guess randomly. Estimate the score of this student in this test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41.25}
In a crown, five diamonds are embedded at five positions, equally spaced around a circle. If there are three different colors of diamonds to choose from, how many different ways are there to embed the diamonds? Assume that the five positions are indistinguishable from each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Given a complex number $z=\frac{4}{5}-\sin \theta +(\cos \theta -\frac{3}{5})i$ is purely imaginary (where $i$ is the imaginary unit), find $\tan \theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{4}{3}}
For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array},  \] for all $n\geq 1$. 

Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1003} - 2}
Initially, the cookie jar had a random but fixed amount of money, M dollars. Doris spent 15% of the initial amount from the cookie jar, Martha spent 10% of the remaining amount after Doris's spending, and John spent 20% of the amount left after Martha's spending. If there were $15 left in the cookie jar after John's spending, how much money, in dollars, was there in the cookie jar at first?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$24.51}
There are 101 nonzero integers written in a circle such that each number is greater than the product of the two numbers that follow it in a clockwise direction. What is the maximum number of negative numbers that could be among these 101 written numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given Gilda starts with a bag of marbles, she gives $30\%$ of them to Pedro, $10\%$ of the remaining to Ebony, $15\%$ of the remaining to Jimmy, and $5\%$ of the remaining to Maria. Calculate the percentage of her original bag of marbles that Gilda has left.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50.87\%}
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled  $A$ . The three vertices adjacent to vertex  $A$  are at heights 10, 11, and 12 above the plane. The distance from vertex  $A$  to the plane can be expressed as  $\tfrac{r-\sqrt{s}}{t}$ , where  $r$ ,  $s$ , and  $t$  are positive integers, and  $r+s+t<1000$ . Find  $r+s+t$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{330}
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:4:5}
It will cost $60 to rent a sailboat and $80 per hour to rent a ski boat on weekdays. During weekends or holidays, the sailboat rental costs $90 and the ski boat rental costs $120 per hour. Ken rented a sailboat while Aldrich rented a ski boat. Additionally, there are fuel costs of $10 per hour for Ken's sailboat rental and $20 per hour for Aldrich's ski boat rental. The company is offering a discount of 10% on the second day of rental. Ken rented the sailboat on a weekday, while Aldrich rented the ski boat during the weekend. How much more did it cost Aldrich to rent the ski boat than Ken to rent a sailboat for three hours a day in two days, accounting for different hourly rates and any applicable discounts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$630}
If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{24}{7}}
A solid wooden rectangular prism measures $3 \times 5 \times 12$. The prism is cut in half by a vertical cut through four vertices, creating two congruent triangular-based prisms. What is the surface area of one of these triangular-based prisms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
In the Cartesian coordinate plane \( xOy \), given a moving line \( l \) that is tangent to the parabola \( \Gamma: y^{2}=4x \) and intersects the hyperbola \( \Omega: x^{2}-y^{2}=1 \) at one point on each of its left and right branches at points \( A \) and \( B \), find the minimum area of \( \triangle OAB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{5}}
In the quadrilateral \(ABCD\), the sides \(AB\) and \(CD\) are parallel, with \(AB\) being twice as long as \(CD\). Point \(S\) is the intersection point of the diagonals of quadrilateral \(ABCD\), and triangles \(ABS\) and \(CDS\) are both equilateral. Point \(E\) lies on the segment \(BS\) such that the angle \(ACE\) measures \(30^{\circ}\).

Determine the ratio of the areas of quadrilateral \(ABCD\) and triangle \(EBC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:1}
Rachelle picks a positive integer \(a\) and writes it next to itself to obtain a new positive integer \(b\). For instance, if \(a=17\), then \(b=1717\). To her surprise, she finds that \(b\) is a multiple of \(a^{2}\). Find the product of all the possible values of \(\frac{b}{a^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Find the sum of the digits of the following product (as a function of \( n \)):
$$
9 \cdot 99 \cdot 9999 \cdot \cdots \cdot \left(10^{2^{n}} - 1\right).
$$

In this product, the number of digits of each factor is twice the number of digits of the previous factor.
(21st United States of America Mathematical Olympiad, 1992)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \cdot 2^n}
For the fractional equation involving x, $\frac{x}{x-2}-2=\frac{m}{x-2}$, find the value of m when the equation has a root. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}