inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the rational map \( \hat{\phi} : \mathbb{P}^2 \setminus \{(1:0:0),(0:1:0),(0:0:1)\} \to \mathbb{P}^2 \) given by \( \phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2}) \) at the point \((1:1:0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0:0:1)}
In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143}
A class went on a camping trip and walked along the road from the school to the base, then immediately returned to the school by the same route. If the distance traveled each day increased by 1 kilometer, and it took 4 days to go and 3 days to return, calculate the distance from the school to the base. Express your answer in kilometers as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Compute the radius of convergence of the power series \( \sum_{n \ge 0} f_n(z) \), where each function \( f_n: \mathbb{C} \rightarrow \mathbb{C} \) is defined by:
\[ f_n(z) = \frac{1}{p_n}[z(1-z)]^{4^n} \]
and \( p_n \) is the largest absolute value of a coefficient in the expansion of \([z(1-z)]^{4^n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Given that $M = Gab^2c^3d^4$ where $a, b, c, d$ are distinct prime numbers, calculate the probability that a pair of numbers randomly chosen from the set of numbers of the form $Ga^ib^jc^kd^l$ (where $0 \leq i \leq 1$, $0 \leq j \leq 2$, $0 \leq k \leq 3$, $0 \leq l \leq 4$) will have a greatest common divisor of $G$ and a least common multiple of $M$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{1785}}
Extract the coefficient of $u^{n-1}$ from the generating function $A = 2^{n-1}[u^{n-1}](\frac{1}{1-u})^n$ to determine the average number of nodes with an even number of children in a plane planted tree with $n$ nodes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{2}}
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2} \frac{(4 \sqrt{2-x}-\sqrt{3 x+2}) d x}{(\sqrt{3 x+2}+4 \sqrt{2-x})(3 x+2)^{2}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{32}\ln5}
There are 4 boxes, and each pair of boxes is weighed once, resulting in weights of 22, 23, 27, 29, and 30 kg for the 5 measurements, respectively. Calculate the difference in weight, in kilograms, between the heaviest box and the lightest one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In how many rearrangements of the numbers1,2,3,4,5,6,7,8,9do the numbers form a \( {hill} \) , that is, the numbers form
 an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in 129876543
 or 258976431?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{254
}
Five friends - Kristina, Nadya, Marina, Liza, and Galya - gather in the park every day after buying ice cream from the shop around the corner. One day, they had a conversation.

Kristina: There were five people in front of me.

Marina: I was the first in line!

Liza: There was no one after me.

Nadya: I was standing next to Marina.

Galya: There was only one person after me.

The girls are friends, so they do not lie to each other. How many people were between Kristina and Nadya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In the hyperbolic plane $\mathbb{H}$, consider two horocycles at points $u$ and $v$ with radii $h$ and $k$, respectively. Find the distance between these horocycles along the geodesic connecting $u$ and $v$ using a Möbius transformation that maps the geodesic to the real axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|k-h|}
What is the tens digit of the five-digit integer N, which is as close as possible to 30000, using the digits 2, 3, 5, 7, and 8 exactly once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find
 the number of5-digit happy integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1110}
6. We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ fields. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board such that the cut goes along the edge $k$. Calculate the last digit of the sum we get when we add all $N(k)$, where $k$ is an inner edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Hannah ran 9 kilometers on Monday at a pace of 6 minutes per kilometer. On Wednesday, she ran 4816 meters at a pace of 5.5 minutes per kilometer and on Friday, she ran 2095 meters at a pace of 7 minutes per kilometer. Calculate how much longer, in minutes, she ran on Monday compared to the combined time for Wednesday and Friday.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.847}
How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
In an isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\) (\(AD > BC\)), the lateral side is 40 cm, and the angle \(\angle BAC\) is \(45^\circ\). Let \(O\) be the center of the circle circumscribed around \(ABCD\). It turns out that point \(O\) lies on the line connecting point \(D\) to the midpoint of side \(AB\). Find the length of the base \(AD\) of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20(\sqrt{6} + \sqrt{2}) \text{ cm}}
What is the maximum number of subsets of  $S = {1, 2, . . . , 2n}$  such that no one is contained in another and no two cover whole  $S$ ?

*Proposed by Fedor Petrov*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \binom{2n}{n-1} }
George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\frac{1}{6}$ each, and the probability of rolling an 8 is $\frac{1}{5}$. What is the expected value of the number shown when this die is rolled? Express your answer as a decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.7667}
The figure below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles. Find the degree measure of one of the larger angles. 

Note: The figure is a nonagon (9-sided polygon).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
How many of the first $1000$ positive integers can be expressed in the form
\[\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion. If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce 11 times before returning to $(2,0)$ for the first time. If $\cot \theta=a-\sqrt{b}$ for positive integers $a$ and $b$, compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{403}
A train 250 m in length crosses a telegraph post in 4 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.5}
Kayla rolls four fair $6$-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
Problem condition

Find the derivative.
$y=6 \arcsin \frac{\sqrt{x}}{2}-\frac{6+x}{2} \cdot \sqrt{x(4-x)}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x^{2}-3}{\sqrt{x(4-x)}}}
A cube with side length \( n \) (\( n \geq 3 \)) is divided into unit cubes by partitions. What is the minimum number of partitions between unit cubes that need to be removed so that it is possible to reach the boundary of the cube from any unit cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-2)^3}
正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 的 9 条棱长都相等, $P$ 是 $C C_{1}$ 的中点, 二面角 $B-A_{1} P-B_{1}=eta$ ，则 $\beta$ 的原始答案为 \frac{\sqrt{m}}{n} 格式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the maximum value of the expression $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$, given that $a, b, c$ are positive real numbers with the constraint $3a + 4b + 5c = 12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \[(2^{a+b}+8)(3^a+3^b) \leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\] holds. Compute the smallest possible value of $128v^2+w^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62208}
Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\left(m-1\right)y+m-6=0$. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
设m是一个给定的正整数，则使得$m ^ { 2 } + n$能整除$n ^ { 2 } + m$的最大正整数n是·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$m ^ { 4 } - m ^ { 2 } + m .$ }
5. There is an unlimited number of chips in six colors. What is the smallest number of chips that need to be arranged in a row so that for any two different colors, there are two adjacent chips of these colors in the row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In a $9 \times 2004$ grid, each integer from 1 to 2004 is filled exactly 9 times. Additionally, within each column, no integer appears more than 3 times. Determine the minimum possible sum of the numbers in the first row of the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670338}
Determine the smallest positive constant \( c \) such that for all sufficiently large \( N \), the inequality \( r(N) \geq cN^{1/2} \) holds, where \( r(N) \) is the least number of quadratic polynomials with non-negative coefficients needed to cover the interval \([1, N]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面, 其球心距为 13, 若作一平面与这二球面相切, 且与圆柱面交成一个椭圆, 则这个椭圆的长轴长与短轴长之和是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the radius of convergence for the infinite series $$ 1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty} a_n (x-3)^n, $$ where the series is not centered at $x=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
What is the interval of validity for the Maclaurin series representation of $\ln(1-8x^5)$, which is given by $\sum_{n=0}^{\infty} \frac{(-1)^{n-1}}{n} (-8x^5)^n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{\sqrt[5]{8}}<x<\frac{1}{\sqrt[5]{8}}}
Given the universe $U=\{(x,y) | x,y \in \mathbb{R}\}$, let set $M=\{(x,y) | \frac{y-3}{x-2} \neq 1\}$ and set $N=\{(x,y) | y \neq x+1\}$. Calculate the complement of $M \cup N$ with respect to $U$, $\complement_U (M \cup N)$. Express your answer as a set of points, including coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{(2,3)\}}
How many ways are there to arrange the numbers $\{ 1,2,3,4,5,6,7,8 \}$ in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
9. Given the ellipse $\frac{x^{2}}{1 \mathrm{~h}}+\frac{y^{2}}{4}=1$ with its left and right foci being $F_{1}$ and $F_{2}$, respectively, point $P$ lies on the line $l: x-\sqrt{3} y+8$ $+2 \sqrt{3}=0$. When $\angle F_{1} P F_{2}$ is maximized, the ratio $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}-1}
Given sets $A = \{|a + 1|, 3, 5\}$ and $B = \{2a + 1, a^2 + 2a, a^2 + 2a - 1\}$, when $A \cap B = \{2, 3\}$, find $A \cup B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-5, 2, 3, 5\}}
Find the radius of convergence of the power series $$ \sum_{n=1}^\infty \frac{x^{n^2}}{2^{n-1} n^n}. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A particular coin has a $\frac{1}{3}$ chance of landing on heads (H), $\frac{1}{3}$ chance of landing on tails (T), and $\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
At Rachelle's school, an A counts as 4 points, a B as 3 points, a C as 2 points, and a D as 1 point. Her GPA for the four classes she is taking is calculated as the total sum of points divided by 4. She is certain to get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a \frac{1}{6} chance of getting an A in English, and a \frac{1}{4} chance of getting a B. In History, she has a \frac{1}{4} chance of getting an A, and a \frac{1}{3} chance of getting a B, independently of her grades in English. What is the probability that Rachelle will achieve a GPA of at least 3.5? Provide your answer in the form of \frac{k}{m} and give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Given the function \(f(x)\) where any point on the curve can be written as \(\left(a, a(1-f'(a)\ln{a})\right)\) for \(a > 0\), find the expression for \(x^{f(x)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{x - 1}}
[ Median to the hypotenuse ] [ Pythagorean Theorem (direct and inverse) ]

Outside the right triangle $A B C$ on its legs $A C$ and $B C$, squares $A C D E$ and $B C F G$ are constructed.

The extension of the altitude $C H$ of triangle $A B C$ intersects the line $D F$ at point $K$. Find $H K$, if the legs are equal to 2 and 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{2\sqrt{13}}}
Tyler has two calculators, both of which initially display zero. The first calculator has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.

A positive integer $n$ is called \textit{ambivalent} if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34776}
Given a 1998-digit number where each pair of adjacent digits forms a two-digit number divisible by 17 or 23, and the last digit of the number is 1. What is the first digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Smith operates a booth where customers arrive according to a Poisson process with a rate of $\lambda = 1$ per minute. He will close the shop the first time $a$ minutes have passed without a customer arriving. Let $X$ be the time until Smith closes his shop. Calculate the expected value $E(X)$. \( \text{Express your answer in terms of } a. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^a - 1}
A math teacher is 176 cm tall. The heights of his grandfather, father, and son are 173 cm, 170 cm, and 182 cm, respectively. Since the son's height is related to the father's height, the teacher uses the method of linear regression analysis to predict his grandson's height as ______ cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{185}
Let \( f(z) \) be an analytic function in the open unit disc \( D \), continuous in the closed unit disc \( \bar{D} \), with \( |f(z)| \leq C \). Suppose \( f(z)/z^n \) can be extended to be analytic in the unit disc. Find a bound for \( |f(z_0)| \) given \( |z_0| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C |z_0|^n}
What is the minimum number of colors needed to paint an annulus divided into 14 regions, such that no two adjacent regions share the same color? Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the system of equations for real numbers $a, b, c, d, e$:
\[ ac = 1, \]
\[ ad + bc = -1, \]
\[ ae + bd = 1, \]
\[ be = -6, \]
find the value of $a + b + c + d + e$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The income and expenditure of a person are in the ratio 10 : 4. If the income of the person is Rs. 19000, and he pays taxes at a rate of 15% on his income, invests 10% of his remaining income on long-term investments, and spends 20% of the leftover amount on short-term investments, find his total savings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11628}
A plane is tiled with squares, each colored with one of 10 colors such that any two adjacent squares have different colors. Determine the minimum number of distinct color pairs that must appear across the edges of the squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $f(x) = x^3 + 3x - 1$ have roots $a, b, c$. Given that \(\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\gcd(m, n) = 1\), find \(100m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3989}
In $\triangle ABC$, let $BC = a$, $CA = b$, and $AB = c$. If $2a^2 + b^2 + c^2 = 4$, find the maximum area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{5}}
In an isosceles right triangle \(\triangle ABC\), point \(E\) is on side \(BC\) such that \(BE = 3CE\). Points \(D\), \(A\), and \(F\) are collinear. In \(\triangle DBE\), let \(h_1\) be the altitude on side \(BE\), and in \(\triangle FEC\), let \(h_2\) be the altitude on side \(EC\). Given that \(3h_1 + h_2 = 3\) cm and the sum of the areas of \(\triangle DBE\) and \(\triangle FEC\) is 6 square centimeters, find the area of \(\triangle ABC\) in square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
We have a five-digit positive integer  $N$ . We select every pair of digits of  $N$  (and keep them in order) to obtain the  $\tbinom52 = 10$  numbers  $33$ ,  $37$ ,  $37$ ,  $37$ ,  $38$ ,  $73$ ,  $77$ ,  $78$ ,  $83$ ,  $87$ .  Find  $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37837}
In a room, there are some people, 2/5 of them are women. If 1/4 of the people are married, the maximum number of women in the room who could be unmarried is 40. How many people are in the room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
There exist complex numbers $z_1, z_2, \dots, z_{10}$ which satisfy the condition: $$|z_k i^k + z_{k+1} i^{k+1}| = |z_{k+1} i^k + z_k i^{k+1}|$$ for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. Given that $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1| + |z_2| + \cdots + |z_{10}|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. How many positive integers $n \leq 1000$ satisfy the condition that $\left[\frac{998}{n}\right]+\left[\frac{999}{n}\right]+\left[\frac{1000}{n}\right]$ is not divisible by 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Determine the number of pairs of positive constants \(a\) and \(b\) such that \(P(a < X < b) = 0.95\), where \(X\) follows a chi-squared distribution with \(n\) degrees of freedom.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{8} \ln 2\)}
Determine the genus of the complete 4-partite graph $K_{4,2,2,2}$, where the vertex set is partitioned into 4 disjoint parts of sizes 4, 2, 2, and 2, and any two vertices are adjacent if and only if they belong to different parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What is the percentage of the flowers that are carnations? Provide your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
There are 10 seats in each of 10 rows of a theatre and all the seats are numbered. Two friends are buying tickets independently. What is the probability that they will occupy adjacent seats? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{55}}