inclusionAI/Ring-lite-rl-data huggingface dataset:
Ann and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens.

Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least $N$ total tokens from the slot machine before Drew receives at least $M=2^{2018}$ total tokens, and Drew wins if he receives $M$ tokens before Ann receives $N$ tokens.

If each person has the same probability of winning, compute the remainder when $N$ is divided by $2018$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In an isosceles triangle, the tangent of the base angle is \( \frac{3}{4} \). Calculate the tangent of the angle formed between the median and the angle bisector drawn to the lateral side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{13}}
Bobbo starts swimming at  $2$  feet/s across a  $100$  foot wide river with a current of  $5$  feet/s.  Bobbo doesn’t know that there is a waterfall  $175$  feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side of the river safely?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find all reals  $a$  such that the sequence  $\{x(n)\}$ ,  $n=0,1,2, \ldots$   that satisfy:  $x(0)=1996$  and  $x_{n+1} = \frac{a}{1+x(n)^2}$  for any natural number  $n$  has a limit as n goes to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ |a| \leq 2 }
直线$\theta =\alpha $与$\rho \cos (\theta -\alpha )=1$的位置关系是________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{垂直}
Using Burnside's lemma, determine the number of distinct necklaces that can be formed with 12 equally spaced stones, using 4 red, 5 green, and 3 blue beads. Consider the group of symmetries given by the dihedral group \(D_{24}\), which includes rotations and reflections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1170}
For a continuous function $g(x):\mathbb{R} \to \mathbb{R}$, let $b_n = \int_{0}^{1/n} g(x) \,dx$. If $\lim\limits_{n\to\infty} b_n = L$, determine the value of $\lim\limits_{n\to\infty} \int_{-1/n}^{0} g(x) \,dx$. \(\text{Express your answer in terms of } L.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L}
Find the least \( n \) such that any subset of \( 1,2,\ldots ,{100} \) with \( n \) elements has 2 elements with a difference of 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
55
}
Given the system of equations:

$$x + y\sqrt{x} = \frac{95}{8}$$
$$y + x\sqrt{y} = \frac{93}{8}$$

where $x, y \in \mathbb{R}$, find the value of $\sqrt{xy}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{4}}
The equation
$$ (x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016) $$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$. Let $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}$ for $k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{2000}.}
Determine for which values of the real positive parameter $b$ the series \(\sum_{n=1}^{\infty }\frac{\cos(n)}{{n^b}+{n^\frac{1}{b}}}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Evaluate the limit: $$\lim_{n\to\infty}{ \int_{0}^{\infty}{\frac{x^n}{1+x^n}}\,dx }$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given real numbers $x, y, z, t \in (0, \pi/2)$ such that
$$
\cos^2(x) + \cos^2(y) + \cos^2(z) + \cos^2(t) = 1.
$$
What is the minimum possible value of
$$
\cot(x) + \cot(y) + \cot(z) + \cot(t)?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Determine the values of $p$ for which the unit ball in $\ell^p(\mathbb{N})$ is weakly compact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Let \( A \) be a 2x2 square matrix with \(|A| \neq 0\) such that \(|A + |A| \text{adj}(A)| = 0\), where \(|A|\) and \(\text{adj}(A)\) denote the determinant and adjugate of matrix \(A\), respectively. Find \(|A - |A| \text{adj}(A)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit as \(x\) approaches infinity: $$ \lim _{x\to \infty }\left(\frac{\arctan \left(1-\cos \left(\frac{1}{x}\right)\right)-\sin \:\left(\frac{1}{2x^2}\right)-e^{-x}}{\ln\left(1+\frac{1}{x^2}\right)-\arctan \left(\frac{1}{x^2}\right)}\right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
To any triangle with side lengths  $a,b,c$  and the corresponding angles  $\alpha, \beta, \gamma$  (measured in radians), the 6-tuple  $(a,b,c,\alpha, \beta, \gamma)$  is assigned. Find the minimum possible number  $n$  of distinct terms in the 6-tuple assigned to a scalene triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single  $1\times1$  square of chocolate and then adds more rows and columns from there. If his current bar has dimensions  $w\times h$  ( $w$  columns and  $h$  rows), then it costs  $w^2$  dollars to add another row and  $h^2$  dollars to add another column. What is the minimum cost to get his chocolate bar to size  $20\times20$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5339}
How many bricks each measuring 30 cm x 12 cm x 10 cm, will be needed to build a wall 6 m x 4 m x 20.5 m?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136,667}
Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:
[list]
[*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
[*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$.
[/list]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the figure shown, \(P\) is a point outside the parallelogram \(ABDC\). It is known that the area of \(\triangle PCD\) is 5 square centimeters, and the area of \(\triangle PAB\) is 11 square centimeters. Find the area of the parallelogram \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the plane $\alpha$, two perpendicular lines are drawn. Line $l$ forms angles of $45^{\circ}$ and $60^{\circ}$ with them. Find the angle that line $l$ makes with the plane $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$99$}
Calculate the number of plane partitions that fit into a 4 x 2 x 3 box using MacMahon's formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
The real number \( a \) makes the equation \( 4^{x} - 4^{-x} = 2 \cos(ax) \) have exactly 2015 solutions. For this \( a \), how many solutions does the equation \( 4^{x} + 4^{-x} = 2 \cos(ax) + 4 \) have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{4030}  distinct roots}
An oil drum in the shape of a cuboid with a height of 5 meters (m) and 20 centimeters (cm) and two sides of the bottom each measuring 6 meters (m), 40 centimeters (cm), and 9 meters (m), is filled with oil. We're going to split it up into several barrels of oil into a oil drum with a volume of 1 cubic meter (m3). Find at least how many oil drums you need.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Determine the order of the general linear group GL(3, 4), where the entries of the matrices are integers modulo 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86016}
Find the number of elements in the intersection $A_7 \cap N_G(H)$, where $G = S_7$ and $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$. Given that $|N_G(H)| = 42$ and $|A_7| = 2520$, use the internal product formula to determine $|A_7 \cap N_G(H)|$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the contour integral \( \int_{\gamma} f(z) \, dz \) where \( f(z)=\frac{1}{(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)} \) and \( \gamma \) is the polygonal path traced by \([0,2,2+2i,2i,0]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( F = \mathbb{F}_{p} \) and \( E = \mathbb{F}_{p^{2}} \) be the degree 2 extension over \( F \). Let \( \tau \) be the nontrivial Galois automorphism of \( E \). Consider a degree one representation \( \phi \) of \( E^{*} \). Define \( \phi^{\tau} : E^* \to \mathbb{C}^{*} \) by \( \phi^{\tau}(x) = \phi(\tau(x)) \) for all \( x \in E^{*} \). Determine the number of representations \( \phi \) such that \( \phi = \phi^{\tau} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p - 1}
Let  $n$  be a positive integer. Jadzia has to write all integers from  $1$  to  $2n-1$  on a board, and she writes each integer  in blue or red color. We say that pair of numbers  $i,j\in \{1,2,3,...,2n-1\}$ , where  $i\leqslant j$ , is  $\textit{good}$  if and only if number of blue numbers among  $i,i+1,...,j$  is odd. Determine, in terms of  $n$ , maximal number of good pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
IMO 1965 Problem A3 The tetrahedron ABCD is divided into two parts by a plane parallel to AB and CD. The distance of the plane from AB is k times its distance from CD. Find the ratio of the volumes of the two parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^2(k+3)}{3k+1}}
Calculate the volume of the region under the surface defined by the function \( f(x,y) = \mathrm{e}^{x^2+y^2} \) and above the annular region in the \((x,y)\) plane defined by \(1 \leq x^2 + y^2 \leq 2\). Use double integrals to find the volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi (e^2 - e)}
Given that \( p \) and \( q \) are real numbers with \( pq = b \) and \( p^{2}q + q^{2}p + p + q = 70 \). If \( c = p^{2} + q^{2} \), find the value of \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Let  $p(x)$  be the monic cubic polynomial with roots  $\sin^2(1^{\circ})$ ,  $\sin^2(3^{\circ})$ , and  $\sin^2(9^{\circ})$ . Suppose that  $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$ , where  $0 <a,b \le 90$  and  $a,b,n$  are positive integers.  What is  $a+b+n$ ?

*Proposed by Andrew Yuan*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
10. Lottery. (From 7th grade, 3 points) It so happened that the Absent-Minded Scholar had only 20 rubles left, and he needed to buy a bus ticket to go home. The ticket costs 45 rubles. Near the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of \( p = 0.1 \), a ticket contains a win of 30 rubles, and there are no other prizes. The Scholar decided to take a risk, as he had nothing to lose. Find the probability that the Absent-Minded Scholar will be able to win enough money to buy a bus ticket.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{419}}
Let $0 < \sigma_i \in \mathbb{R}$ ($i=1 \dots d$). Determine the best constant $c$ such that the following inequality holds:

$$ c \sum_{i=1}^d (\sigma_i-1)^2 \le \sqrt{\sum_{i=1}^d (\sigma_i^2-1)^2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{\sqrt{d}}\)}
Evaluate the limit for a Gaussian measure $\mu$ on $\mathbb{R}^m$:
$$\lim_{n \to \infty} \frac{1}{\mu(B_n) \sup_{x \in B_n}\lVert x \rVert^2} \int_{B_n} \lVert x \rVert^4 d\mu(x),$$
where $B_n$ is the closed ball at the origin with radius $1/n$. \( \mu \) is a Gaussian measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given an AR(1) process with correlation parameter \(\rho\), determine the maximum possible correlation between its elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the function \( u \in C^1[0,1] \) that minimizes the functional \( J(y) \), where
\[ J(y) = \frac{1}{2}\int_0^1 (y'^2+y^2) \, \mathrm{d}x + \frac{1}{2}[y^2(0)+y^2(1)] - 2y(0). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-x}}
Evaluate the integral: \[ \int_{-2\pi}^{2\pi}(1−u_0(x))\sin(x/2)(\delta(x + \pi) + \delta(x−\pi))\mathrm{d}x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let G, O, D, I, and T be digits that satisfy the following equation:
$$
\begin{array}{c}
&G&O&G&O\\+
&D&I&D&I\\
\hline
G&O&D&O&T\end{array}
$$
(Note that G and D cannot be $0$, and that the five variables are not necessarily different.)

Compute the value of GODOT.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10908}
Given that the training scores X of high school teachers in Harbin city follow a normal distribution N(85, 9) and P(80<X≤85)=0.35, calculate the probability that a randomly selected teacher from Harbin city has a training score greater than 90. Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.15}
Let $T: \mathbb{C}^{11} \to \mathbb{C}^{11}$ be a linear transformation. Given that the dimension of the kernel of $T$ is 4, the dimension of the kernel of $T^3$ is 9, and the dimension of the kernel of $T^4$ is 11, find the dimension of the kernel of $T^2$. Use the rank-nullity theorem and the properties of nilpotent operators to determine your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33725}
In a game, \( N \) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \( m \) be the maximum possible number such that it is possible to win the game by writing down \( m \). Find the smallest possible value of \( N \) for which it is possible to win the game by writing down \( m \) in a room of \( N \) people.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Jeffrey writes the numbers 1 and \( 100000000 = 10^8 \) on the blackboard. Every minute, if \( x \) and \( y \) are on the board, Jeffrey replaces them with
\[ 
\frac{x + y}{2} \text{ and } 2\left(\frac{1}{x} + \frac{1}{y}\right)^{-1}. 
\]
After 2017 minutes, the two numbers are \( a \) and \( b \). Find \( \min(a, b) \) to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
A flagpole is 12 feet tall. It breaks, folding over in half, such that what was the tip of the flagpole is now dangling at a certain height above the ground. The flagpole broke 7 feet from the base. How high is the tip of the flagpole from the ground after it broke?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Georg is putting his $250$ stamps in a new album. On the first page, he places one stamp, and then on every subsequent page, he places either the same number of stamps or twice as many as on the preceding page. In this way, he ends up using precisely all $250$ stamps. How few pages are sufficient for him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Calculate the expectation \( \mathbb{E}\left(\int_0^1 B(t) \, dB(t) \cdot \int_0^1 g(t) \, dB(t)\right) \) where \( B(t) \) is a Brownian motion and \( g(t) \) is a deterministic function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a convex 10-gon \(A_{1} A_{2} \ldots A_{10}\), all sides and all diagonals connecting vertices skipping one (i.e., \(A_{1} A_{3}, A_{2} A_{4},\) etc.) are drawn, except for the side \(A_{1} A_{10}\) and the diagonals \(A_{1} A_{9}\), \(A_{2} A_{10}\).

A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-intersecting broken line (i.e., a line such that no two nonconsecutive segments share a common point) with endpoints \(A_{1}\) and \(A_{10}\), where each segment coincides with one of the drawn sides or diagonals. Determine the number of such paths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
How many of the 2401 smallest positive integers written in base 7 use 3 or 6 (or both) as a digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1776}
Evaluate the limit: \[ \lim_{n \to \infty} \left[ \int_{-\infty}^{x \ln n} \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} \, dy \right]^n \] where \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the ratio \( \frac{I_1}{I_2} \) where:

\[ I_1 = \int_0^1 \frac{dt}{\sqrt{1-t^4}} \]
\[ I_2 = \int_0^1 \frac{dt}{\sqrt{1+t^4}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
一个六面体的各个面和一个正八面体的各个面都是边长为 $a$ 的正三角形, 这样两个多面体的内切球的半径之比是一个既约分数 $\frac{m}{n}$ 。那么, 积 $m \cdot n$ 是
$\qquad$ $\cdot$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given $A=\{y\in \mathbb{N}|y=-x^{2}+4x,x\in \mathbb{Z}\}$ and $B=\{x|\ln x > 1\}$, find the intersection of set $A$ and the complement of set $B$ in $\mathbb{R}$. Express your answer as a set of numbers using standard set notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Determine the dimension of \(\mathbb{Q}(t)\) as a \(\mathbb{Q}(t^2)\)-vector space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the minimum value of $a$ such that the function $\theta \mapsto \frac{\sin \theta}{a + \cos \theta}$ is bounded by $\pm 2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{5}}{2}\)}
给定整数$n\geqslant3$。求最小的正整数$k$，使得存在一个$k$元集合$A$和$n$个两两不同的实数$x_{1}, x_{2}, \cdots, x_{n}$，满足$x_{1}+x_{2}, x_{2}+x_{3}, \cdots, x_{n-1}+x_{n}, x_{n}+x_{1}$均属于$A$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a building with \( n \) floors and two staircases going from the first to the last floor, there is a door on each intermediate landing between every two floors. The doors separate the floors (you can access the floor from the staircase, even if the door is locked). The building superintendent decided that too many open doors are undesirable and locked exactly half of the doors, choosing the doors randomly. What is the probability that it is possible to ascend from the first floor to the last floor, passing only through open doors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^{n-1}}{\binom{2(n-1)}{n-1}}}
The figure "nedoslon" attacks one square diagonally. What is the maximum number of non-attacking "nedoslons" that can be arranged on a $9 \times 9$ chessboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Consider the first $n$ natural numbers and randomly pick $m$ numbers out of these. Define S as the set of these $m$ numbers, having density $d=m/n$. Consider all arithmetic progressions of length 3 in S. What is the expectation of the proportion of these triples that crosses $n/2$? That is, the progressions that start at < $n/2$ and have endpoint > $n/2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
There is a right triangle $\triangle ABC$ where $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ such that $\angle ACX = \angle XCY = \angle YCZ = \angle ZCB$ and $BZ = 2AX$. The smallest angle of $\triangle ABC$ is $\frac{a}{b}$ degrees, where $a$ and $b$ are positive integers such that $\gcd(a, b) = 1$. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{277}
Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, find the length $B'C'$. The original answer is in the form $k+\sqrt{m(r^2-1)}$, please give the value of $k+m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a $20 \times 19$ grid, use $3 \times 1$ dominoes to cover the grid without overlapping and without common points. Find the maximum possible number of dominoes needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
In the ring \(R=\mathbb{Z}_7[x]\), consider the element \(b=(x+2)(x+5)\). Determine the number of elements \(a\) in \(R\) such that the equation \(at=b\) has a solution \(t\in R\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the integral \( \int_0^1 \frac{x-1}{\log x}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 2}
Let \( x < 0.1 \) be a positive real number. Consider the series \( 4 + 4x + 4x^2 + 4x^3 + \ldots \), and the series \( 4 + 44x + 444x^2 + 4444x^3 + \ldots \). Suppose that the sum of the second series is four times the sum of the first series. Compute \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{40}}