inclusionAI/Ring-lite-rl-data huggingface dataset:
在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形 $A B C D$ 所在平面上一点, 满足
$$
\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\overrightarrow{0}
$$
设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$ 的面积, 则 $\frac{t}{s}=$ $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2358}
Given the sequence $\left\{a_{n}\right\}$ with the partial sum $S_{n}=2 a_{n}-1$ for $n=1,2,\cdots$, and the sequence $\left\{b_{n}\right\}$ that satisfies $b_{1}=3$ and $b_{k+1}=a_{k}+b_{k}$ for $k=1,2,\cdots$, find the partial sum of the sequence $\left\{b_{n}\right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n + 2n - 1}
Let $p,$ $q,$ $r,$ $x,$ $y,$ and $z$ be positive real numbers such that $p + q + r = 2$ and $x + y + z = 1$. Find the maximum value of:
\[\frac{1}{p + q} + \frac{1}{p + r} + \frac{1}{q + r} + \frac{1}{x + y} + \frac{1}{x + z} + \frac{1}{y + z}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{4}}
In quadrilateral ABCD, AB = √2, BC = √3, CD = 1, ∠ABC = 75°, ∠BCD = 120°. Find the measure of ∠CDA. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A train of a certain length running at the speed of 54 km/hr takes 16.13204276991174 seconds to cross a bridge 132 m in length. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109.98}
Find the area of the closed figure formed by the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} - \frac{\pi}{3}}
Let $f(n)$ be the sum of the squares of the digits of the positive integer $n$ (in decimal notation). For example, $f(123) = 1^2 + 2^2 + 3^2$. Define $f_1(n) = f(n)$, and $f_{k+1}(n) = f[f_k(n)]$ (where $k=1,2,3,\ldots$). Calculate the value of $f_{2007}(2007)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{145}
Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Suppose $G$ is a group with exactly $12$ elements of order $4$. How many elements of order $2$ does $G$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \(\Gamma\) denote the circumcircle of triangle \(ABC\). Point \(D\) is on \(\overline{AB}\) such that \(\overline{CD}\) bisects \(\angle ACB\). Points \(P\) and \(Q\) are on \(\Gamma\) such that \(\overline{PQ}\) passes through \(D\) and is perpendicular to \(\overline{CD}\). Compute \(PQ\), given that \(BC = 20\), \(CA = 80\), and \(AB = 65\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{745}}
In how many ways can a committee of three people be formed if the members are to be chosen from four married couples?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? (For example, Raashan and Ted may each decide to give $1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0$, Sylvia will have $2$, and Ted will have $1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $1$ to, and the holdings will be the same at the end of the second round.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
A box contains some tablets of medicine A and 14 tablets of medicine B. The least number of tablets that should be taken from the box to ensure that at least two tablets of each kind are among the extracted is 16. How many tablets of medicine A are in the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In Mrs. Evans' freshmen economics class, 40 students are enrolled. Some students answered question 1 correctly, 29 answered question 2 correctly, and 10 did not take the test. If 29 students answered both questions correctly, how many students answered question 1 correctly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
For positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the smallest positive integer $m$ such that
\[
r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Tim came second in math. When his mother asked him how much he had scored, he answered that he got the sum of the first 'n' even numbers, where 'n' is a square number. His mother immediately worked out the answer. What is the value of 'n'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a sequence $\{a_{n}\}$ with $a_{1}=111$, $a_{2}=217$, and for $n\geqslant 3$, $a_{n}=a_{n-2}-\frac{n-p}{a_{n-1}}$, find the value of the constant $p$ when the maximum number of terms $m$ is $220$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{110}{109}}
For which $n$ can values of the variables $x_{1}, x_{2}, \ldots, x_{n}$ be chosen from the set $\{+1, -1\}$ such that the equation $x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + \ldots + x_{n-1}x_{n} + x_{n}x_{1}=0$ is satisfied?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is divisible by 4}}
A Battleship game board measures $10 \times 10$. What is the maximum number of $1 \times 4$ ships that can be placed on it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
5. In an isosceles trapezoid $A B C D$ with lateral sides $A B$ and $C D$, the lengths of which are 10, perpendiculars $B H$ and $D K$ are drawn from vertices $B$ and $D$ to the diagonal $A C$. It is known that the bases of the perpendiculars lie on segment $A C$ and $A H: A K: A C=5: 14: 15$. Find the area of trapezoid $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
For positive reals \( p \) and \( q \), define the remainder when \( p \) is divided by \( q \) as the smallest nonnegative real \( r \) such that \( \frac{p - r}{q} \) is an integer. For an ordered pair \( (a, b) \) of positive integers, let \( r_1 \) and \( r_2 \) be the remainder when \( a \sqrt{2} + b \sqrt{3} \) is divided by \( \sqrt{2} \) and \( \sqrt{3} \) respectively. Find the number of pairs \( (a, b) \) such that \( a, b \leq 20 \) and \( r_1 + r_2 = \sqrt{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In triangle $\triangle ABC$, $\angle C=50^{\circ}$. With point $C$ as the center, draw an arc with a suitable length as the radius, intersecting $AC$ at point $M$ and $BC$ at point $N$. With $M$ and $N$ as centers, draw arcs with lengths greater than $\frac{1}{2}MN$ as the radius. These two arcs intersect at point $D$. Connect $CD$. Find the degree of $\angle ACD$. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Consider the set \( S = \{1, 2, \ldots, 30\} \). There are 10 subsets, \( A_1, A_2, \ldots, A_{10} \), each containing 3 elements, where any two subsets \( A_i \) and \( A_j \) have at least one element in common. Define \( n_i \) as the number of subsets \( A_j \) to which element \( i \) belongs. Determine the smallest possible value of \( \max (n_i) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A circle with radius $r$ is tangent to sides $AB, AD$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. The area of the rectangle, in terms of $r$, is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8r^2$}
Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\{1,2, \ldots, k\}$ is a multiple of 11 can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Find the minimum value of the function \( u(x, y) = x^{2} + \frac{81}{x^{2}} - 2xy + \frac{18}{x} \sqrt{2 - y^{2}} \) where \( x \) and \( y \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the sides $a, b, c, d$ of a trapezoid, how long is the line segment that joins the midpoints of the parallel sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a + b}{2}}
Let $M=\{x|0 < x < 4\}$ and $N=\{x|\frac{1}{3}\leqslant x\leqslant 5\}$. Find the intersection $M \cap N$ and express your answer as a set definition in the format $\{x|\text{condition}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{x|\frac{1}{3}\leqslant x < 4\}}
Given the lines $l_{1}$: $\left(m+3\right)x+5y=5-3m$ and $l_{2}$: $2x+\left(m+6\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, then the value of $m$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
Two trains, 431 meters and 365 meters in length respectively, are running in opposite directions, one at the rate of 120 km/h and the other at the rate of 105 km/h. In what time will they be completely clear of each other from the moment they meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.736}
If the graph of the inverse proportion function $y=\frac{2}{x}$ passes through points $A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$, $C(x_{3}, y_{3})$, and $x_{1} \lt x_{2} \lt 0 \lt x_{3}$, then the relationship between $y_{1}$, $y_{2}$, $y_{3}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_{2} < y_{1} < y_{3}}
Point \( I \) is the center of the inscribed circle of triangle \( ABC \), point \( O \) is the center of the escribed circle that touches side \( AC \), and line segments \( AC \) and \( OI \) intersect at point \( K \).

It is known that \( OI = 50, IK = 18, \) and \( AK = 24 \). Find the length of the angle bisector of angle \( B \) in triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{576}{7}}
Farm tax is levied on the 60% of the cultivated land. The tax department collected a certain amount through the farm tax from the village of Mr. Willam. Mr. Willam paid only $500 as farm tax. The percentage of total land of Mr. Willam over the total taxable land of the village is 20.833333333333332%. How much money did the tax department collect from the village?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$2400}
Determine the limiting value of the following expression as $\beta$ approaches infinity:

$$\lim_{\beta\to+\infty}\frac{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}(t-1)dt}{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}dt}$$

where $I_0(x)$ represents the modified Bessel function of the first kind.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer tasks receive fewer candies, possibly receiving none. Determine the smallest value of $N$ for which this distribution is possible, regardless of the number of tasks in the Olympiad and the students' successes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{41}}
四个半径为 1 的球两两相切，则它们的外切正四面体的棱长为多少？答案的形式为k(1+\sqrt{m})，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given positive numbers $x$, $y$, and $z$ that satisfy the equation $x^2 + y^2 + z^2 = 2xyz + 1$, find the maximum value of the expression:
$$P = (x - 2yz)(y - 2zx)(z - 2xy).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
Find the number of all $6$-digit natural numbers such that the sum of their digits is $10$ and each of the digits $0, 1, 2, 3$ occurs at least once in them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
Patrick has been saving money to buy a bicycle that costs $150. He saved half the price, but then lent $50 to his friend Alice at a 5% annual interest rate and $30 to his friend Bob at a 7% annual interest rate. If Alice repays the loan after 8 months and Bob after 6 months, how much money does Patrick have in total, including the interest accrued on both loans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$157.72}
Ali turned 8 years old this year and is now 3 years older than Yusaf. Umar is twice Yusaf's age, and Aisha is half a year younger than the average age of Ali and Umar. If Aisha reached a milestone age last month, how old is Aisha?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If the price of a book is first decreased by 40% and then increased by 35%, what will be the net change in the price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19\%}
An integer $n$ is chosen uniformly at random from the set $\{1,2,3, \ldots, 2023!\}$. Compute the probability that $$\operatorname{gcd}\left(n^{n}+50, n+1\right)=1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{265}{357}}
Given $\sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{4}$ with $\theta \in \left(-\frac{3\pi}{2}, -\pi\right)$, find the value of $\cos\left(\theta + \frac{7\pi}{12}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\sqrt{15} + \sqrt{3}}{8}}
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Calculate the maximum slope of the line $OQ$. Express your answer as a single numerical value, including the square root symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
On an island, there live knights, liars, and yes-men; each knows who is who among them. In a row, 2018 island inhabitants were asked the question: "Are there more knights than liars on the island?" Each inhabitant answered either "Yes" or "No" in turn such that everyone else could hear their response. Knights always tell the truth, liars always lie. Each yes-man answered according to the majority of responses given before their turn, or gave either response if there was a tie. It turned out that exactly 1009 answered "Yes." What is the maximum number of yes-men that could be among the island inhabitants?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remaining purple shirts are also hanging consecutively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
On the right pan of a balance scale lies a weight with a mass of 11111 g. The scale operator successively places weights on the pans, the first of which has a mass of 1 g, and each subsequent one is twice as heavy as the previous one. At some point, the scales are balanced. On which pan is the 16 g weight placed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The 16-gram weight is on the left pan}
Given the sequence $$a_1=\sqrt{2}, \quad a_{n+1}=\sqrt{2}\left(a_n+\frac {1}{a_n}\right),$$ compute $$\lim_{n\to \infty}\frac{a_n}{(\sqrt{2})^n}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If the line $y=2x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0, b \gt 0\right)$ do not have any common points, then the range of eccentricity of the hyperbola is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \sqrt{5}]}
Given that the inequality $2x-m\leqslant 3$ holds, a sufficient and necessary condition is $-5 \lt x \lt 4$. The range of real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{m\ |\ m\geq 5\}}
11. A Trip for Milk. Anton went to a dairy store. He had no money, but he had empty bottles - six one-liter bottles (worth 20 kopecks each) and six half-liter bottles (worth 15 kopecks each). In the store, there was milk sold by the liter for 22 kopecks. What is the maximum amount of milk he could bring home? He had no other containers besides the empty bottles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If a function $g$ satisfies $g(x) = g(x+3)$ for all $x \in \mathbb{R}$, and it is given that $\int_2^4 g(x) dx = 5$ and $\int_{-3}^{-1} g(x) dx = -3$, calculate the value of $\int_1^{10} g(x) dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Problem: Given $f \in C^{(1)} [0,1]$, with $f(0) = a$ and $\int_{0}^{1} f(x) \, dx = 0$, determine the minimum value of $\int_{0}^{1} (f'(x))^2 \, dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3a^2}
Given that there are 5 points in the plane no three of which are collinear, draw all the segments whose vertices are these points, and determine the minimum number of new points made by the intersection of the drawn segments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if 
\[k = \sum_{i=2}^n a_i F_i,\]
where $a_i \in \{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i \ge 2$). This representation is said to be \textit{minimal} if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1596}
$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
How many non-trivial ring homomorphisms are there from the ring $Z_{20}$ to the ring $Z_{44}$, considering them as rings modulo 20 and 44, respectively?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are 5 volunteers to be divided into 3 groups to serve at three different venues for the Youth Olympic Games. Two of the groups will have 2 people each, and one group will have 1 person. How many different allocation schemes are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
What is the value of  
$$3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
The answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
In a freight yard, there are 100 shipping containers. Each container carries 30 crates. Each crate is filled with 15 boxes, and each box is filled with 10 washing machines. There are 6 workers, who work simultaneously to remove a certain number of washing machines from each box. If each worker removes 4 washing machines from each box they manage, how many washing machines were removed in total by all workers? Note that all workers remove the same number of washing machines from each box.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180000}
设 a,b>0，满足：关于 x 的方程 \sqrt{|x|}+\sqrt{|x+a|}=b 恰有三个不同的实数解 x_1, x_2, x_3，且 x_1<x_2<x_3=b，则 a+b 的值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6956}
How many ordered pairs of real numbers  $(x, y)$  are there such that  $x^2+y^2 = 200$  and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A tank has 3 inlet pipes A, B, and C with flow rates of 8 liters/min, 12 liters/min, and 6 liters/min respectively. The tank also has 2 outlet pipes, X and Y, with flow rates of 20 liters/min and 15 liters/min respectively. Inlet pipe A and outlet pipe X are open for the entire time, while inlet pipes B and C are open for the first 3 hours and 4 hours, respectively. Outlet pipe Y is open for the last 5 hours. At the end of 7 hours, the tank is full. Find the capacity of the tank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5940}
Given that $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > b > 0)$, a circle with the origin as its center and the semi-focal distance as its radius intersects the right branch of the hyperbola at points $A$ and $B$, and $\triangle F_{1}AB$ is an equilateral triangle, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} + 1}
A bug starts at a vertex of a grid made of equilateral triangles with a side length of $1$. At each step, the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. Find the probability that after $5$ moves, the bug will never have been more than $1$ unit away from the starting position. 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{121}