inclusionAI/Ring-lite-rl-data huggingface dataset:
Initially, there are two knights on a $2018 \times 2018$ board: a red knight in the bottom left corner and a blue knight in the bottom right corner. Kolya and Sasha take turns moving; Kolya starts. On each turn, a player moves their knight (Kolya moves the red knight, and Sasha moves the blue knight), shifting it simultaneously by 20 cells in one direction and by 17 cells in the other direction. A knight cannot move to a cell already occupied by the other knight. It is forbidden to create a position that has already occurred in the game (positions are considered the same if the red knight is on the same cell and the blue knight is also on the same cell). The player who cannot make a move loses. Who will win with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Sasha}
Determine the values of $a \in \mathbb{R}$ for which the function \[ f_n(x,y,z) = \frac{\arctan^a (|x|+|y|)e^z}{n^4+4x^2+y^2} \] is in $L^1(D)$ for all $n \geq 1$, where $D = \{(x,y,z) : 1 < x^2 + y^2 + z < 2\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, \infty)}
Evaluate the integral: \[ \frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }\frac{k}{2}\sqrt{\frac{\pi}{2}}e^{-2|k-\pi|}dk \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
$11分制乒乓球比赛，每赢一球得1分，当某局打成10:10平后，每球交换发球权，先多得2分的一方获胜，该局比赛结束.甲、乙两位同学进行单打比赛，假设甲发球时甲得分的概率为0.5，乙发球时甲得分的概率为0.4，各球的结果相互独立.在某局双方10:10平后,甲先发球，两人又打了i个球该局比赛结束.$
$求事件“X=4且甲获胜”的概率.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$0.1$}
If the line $l: x-my+9=0$ intersects the circle $C: x^{2}+y^{2}+2x-24=0$ and forms a line segment of length $6$, then the value of the real number $m$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\sqrt{3}}
Find all values of the parameter \( a \) for which the equation \( \left|x^{3}-a^{3}\right|=x-a \) has three distinct roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-2}{\sqrt{3}} < a < \frac{-1}{\sqrt{3}}}
Given the system of differential equations:
\[\frac{\mathrm{d}x}{\mathrm{d}t}=ax+by\]
\[\frac{\mathrm{d}y}{\mathrm{d}t}=cx+dy\]
where the solution \((x(t), y(t))\) lies on a straight line, find the value of \(y(10)\) given that \(y(0)=5\), \(x(0)=6\), and \(x(10)=2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{3}}
Calculate the expected value of the following game: You roll a fair six-sided die up to 60 times and receive an amount in dollars equal to the last number rolled. You can choose to stop rolling at any time. What is the expected value of this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday? Express your answer as a single year in the format YYYY.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
A geometric sequence of positive integers has its first term as 5 and its fourth term as 480. What is the second term of the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
7.5. On a chessboard, the centers of some cells are marked in such a way that no triangle with marked vertices is a right triangle. What is the maximum number of points that could have been marked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find an element $m \in U(16)$ such that $|m| = 4$ and $|\langle m \rangle \cap \langle 3 \rangle| = 2$. The operation is defined as $m*n = mn \pmod{16}$. Determine if $m$ is unique.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $M$ and $N$ be smooth manifolds and let $F: M \to N$ be a smooth embedding. Consider the differential structure $DS_1$ on $F(M)$ induced by an atlas $\Sigma = \{(U_{\alpha}, \varphi_{\alpha})\}$ on $M$. Also, consider the differential structure $DS_2$ on $F(M)$ as a regular submanifold of $N$. Must we have $DS_1 = DS_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{DS_1 = DS_2}
Evaluate the integral \( \int_{-1}^1 x \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
In right triangle ABC, ∠C=90° and AB=2BC. Find the value of cos A. Express your answer as a fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
Evaluate the integral \( \int (10x^2 + 3y^3 - 4z^2 + 8) \delta(2\vec{r}) \, d^3\vec{r} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine whether the limit of the sequence \( \{u_n\}_{n\ge 0} \), defined by the recurrence relation \( u_{n+1} = -\frac{1}{e} + (n+1)u_n \) with the initial condition \( u_0 = 1 - \frac{1}{e} \), can be found directly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the sequence \( \frac{e^n \cdot n!}{n^n} \) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A manufacturer of cameras offers a two-year warranty for each camera, which consists of 8 components. Each component has a 3% chance of breaking during the warranty period. If a component breaks, it costs $50 to repair or replace it. The company plans to sell 800 cameras at launch. Determine the price at which each camera should be sold so that there is a 98% probability of making a profit of at least $120,000. Use the normal approximation to solve this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163.75}
Find the best asymptotic expansion for \(x \to 0\) of the expression:

\[ \sqrt{1-2x+x^2+o(x^3)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - x + o(x^3)}
There are $5$ apples, $8$ pears, and $11$ oranges in a basket. Every day Alice eats two different fruits. Can it happen that some day there are the same number of apples, pears and oranges remaining in the basket?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
The first digit on the left of a six-digit number is 1. If this digit is moved to the last place, the resulting number is 64 times greater than the original number. Find the original number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142857}
Which of the following does not divide $n^{2225}-n^{2005}$ for every integer value of $n$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 5 black squares, can be drawn on the checkerboard?

[asy]
draw((0,0)--(8,0)--(8,8)--(0,8)--cycle);
draw((1,8)--(1,0));
draw((7,8)--(7,0));
draw((6,8)--(6,0));
draw((5,8)--(5,0));
draw((4,8)--(4,0));
draw((3,8)--(3,0));
draw((2,8)--(2,0));
draw((0,1)--(8,1));
draw((0,2)--(8,2));
draw((0,3)--(8,3));
draw((0,4)--(8,4));
draw((0,5)--(8,5));
draw((0,6)--(8,6));
draw((0,7)--(8,7));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black);
fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,black);
fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black);
fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black);
fill((4,2)--(5,2)--(5,3)--(4,3)--cycle,black);
fill((6,2)--(7,2)--(7,3)--(6,3)--cycle,black);
fill((0,4)--(1,4)--(1,5)--(0,5)--cycle,black);
fill((2,4)--(3,4)--(3,5)--(2,5)--cycle,black);
fill((4,4)--(5,4)--(5,5)--(4,5)--cycle,black);
fill((6,4)--(7,4)--(7,5)--(6,5)--cycle,black);
fill((0,6)--(1,6)--(1,7)--(0,7)--cycle,black);
fill((2,6)--(3,6)--(3,7)--(2,7)--cycle,black);
fill((4,6)--(5,6)--(5,7)--(4,7)--cycle,black);
fill((6,6)--(7,6)--(7,7)--(6,7)--cycle,black);
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,black);
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black);
fill((5,1)--(6,1)--(6,2)--(5,2)--cycle,black);
fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,black);
fill((1,3)--(2,3)--(2,4)--(1,4)--cycle,black);
fill((3,3)--(4,3)--(4,4)--(3,4)--cycle,black);
fill((5,3)--(6,3)--(6,4)--(5,4)--cycle,black);
fill((7,3)--(8,3)--(8,4)--(7,4)--cycle,black);
fill((1,5)--(2,5)--(2,6)--(1,6)--cycle,black);
fill((3,5)--(4,5)--(4,6)--(3,6)--cycle,black);
fill((5,5)--(6,5)--(6,6)--(5,6)--cycle,black);
fill((7,5)--(8,5)--(8,6)--(7,6)--cycle,black);
fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black);
fill((3,7)--(4,7)--(4,8)--(3,8)--cycle,black);
fill((5,7)--(6,7)--(6,8)--(5,8)--cycle,black);
fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);

[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Evaluate the double integral in polar coordinates to find the volume of the 3D-body defined by \( z = y + \sqrt{4 - x^2} \) over the region where \( x \leq 0 \) and within the circle of radius 2:
\[
\int_{\pi}^{2\pi}\int_{0}^{2} \left( r\sin \theta + \sqrt{4-r^2\cos^2\theta} \right) r\,drd\theta
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{3}}
Under the mapping \( w = \frac{1}{z + 2 + 3i} \), where \( z = x + iy \), the image of the line \( x + y = 1 \) is a circle minus a point \( w_0 \). Given that the circle is \( (x - \frac{1}{12})^2 + (y + \frac{1}{12})^2 = \frac{1}{72} \), find the point \( w_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
An omino is a 1-by-1 square or a 1-by-2 horizontal rectangle. An omino tiling of a region of the plane is a way of covering it (and only it) by ominoes. How many omino tilings are there of a 2-by-10 horizontal rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7921}
The operation $ \diamond $ is defined for positive integers $a$ and $b$ such that $a \diamond b = a^2 - b$. Determine how many positive integers $x$ exist such that $20 \diamond x$ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Find the limiting value of the sequence \( \sum_{k=1}^{n}\frac{1}{\binom{n}{k}} \) as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
a searchlight on top of the watchtower makes 2 revolutions per minute . what is the probability that a man appearing near the tower will stay in the dark for at least 10 seconds ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
Consider a simple random walk $(X_n)$ starting from $X_0 = 0$, where on each step it moves up by one with probability $p$ and down by one with probability $q = 1 - p$. What is the conditional probability $P(X_4 = 4 | X_8 = 6)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
14th Mexico 2000 Problem B1 Given positive integers a, b (neither a multiple of 5) we construct a sequence as follows: a 1 = 5, a n+1 = a a n + b. What is the largest number of primes that can be obtained before the first composite member of the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit of the integral as \( n \) approaches infinity: \[ \lim\limits_{n \to \infty} \int_0^n \frac{1}{1 + n^2 \cos^2 x} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If $R$ and $S$ are graded rings, and $h_n:R_n\longrightarrow S_n$ is an isomorphism of abelian groups for each $n$, is $h$ a ring isomorphism in the natural way? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
A ball of radius \(R\) and mass \(m\) is magically put inside a thin shell of the same mass and radius \(2R\). The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface? The original answer is in \(\frac{a}{b}\) format, please give the value of a + b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered $1,2,3, \ldots, 20$. Then, starting with the player with the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends. Given that Yaiza lost and 5 cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player's hand does not matter.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
If the following triangular array of numbers is continued using the pattern established, how many numbers (not how many digits) would there be in the $100^{th}$ row? As an example, the $5^{th}$ row has $11$ numbers. Use exponent notation to express your answer.
$$
\begin{align*}  &1 \\  &2 \\ 3\quad &4\quad  5\quad  \\ 6\quad   7\quad  &8\quad  9\quad  10\quad  \\ 11\quad  12\quad  13\quad  14\quad  15\quad  &16\quad  17\quad  18\quad  19\quad  20\quad  21\quad  \\ 22\quad  23\quad  24\quad  25\quad  26\quad  27\quad  28\quad  29\quad  30\quad  31\quad  &32\quad  33\quad  34 \quad 35\quad  36\quad  37\quad  38\quad  39\quad  40\quad  41\quad  42\quad  \\ \cdot \quad &\cdot\quad  \cdot\quad \\ \end{align*}
$$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2^{100}-1}{3}$}
Determine the number of zeros of the polynomial \( z^9 + z^5 - 8z^3 + 2z + 1 \) that lie in the annular region \( 1 < |z| < 2 \). Use Rouche's theorem to justify your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
From the same number of squares with sides 1, 2, and 3, form a square of the smallest possible size.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
John has two jobs. From job A, he makes $60 a week and receives a raise increasing his weekly pay to $78. He also gets a quarterly bonus of $50 that remains the same. From job B, he makes $100 a week and receives two raises: the first one increases his weekly pay by $10, and the second one by $5, giving him a total of $115 a week. Job B also provides a biannual bonus of $200, but after the raises, his biannual bonus is increased by 10%. Additionally, John has weekly expenses of $30 that remain constant. What is the net percentage change in John's effective quarterly income after considering his raises, bonuses, and expenses?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.86\%}
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac{m}{n}$, 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{919}
Determine whether the sequence \( b_n = \int_0^n \int_0^{\frac{1}{n}} \int_0^1 \int_0^{\frac{1}{n}} \log^2\left[(x-t)^2+(y-s)^2\right] \, dx \, dy \, dt \, ds \) converges or diverges as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If the base of a rectangle is increased by $10\%$ and the area is unchanged, then the altitude is decreased by:




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$9\frac{1}{11}\%$}
Determine the minimum number of roots in the interval \([0, 2\pi)\) for the equation \( \sin(k_{0} x) + A_{1} \cdot \sin(k_{1} x) + A_{2} \cdot \sin(k_{2} x) = 0 \), where \( k_{0}<k_{1}<k_{2} \) are natural numbers and \( A_{1}, A_{2} \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the figure, ABCDE is a regular pentagon. Given that $AD = AF$ and $\angle DAF = 56^\circ$, find the measure of angle $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Rosencrantz plays $n \leq 2015$ games of question, and ends up with a win rate (i.e. $\frac{\# \text { of games won }}{\# \text { of games played }}$ ) of $k$. Guildenstern has also played several games, and has a win rate less than $k$. He realizes that if, after playing some more games, his win rate becomes higher than $k$, then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2015}}
20. Combine a regular hexagon with a side length of 1 and an equilateral triangle with a side length of 1, requiring no overlap, and the sides to be completely coincident, then the number of sides of the new shape is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A cylindrical cup containing some water is tilted 45 degrees from the
 vertical. The point on the surface of the water closest to the bottom of the
 cup is 6 units away. The point on the surface of the water farthest from the
 bottom of the cup is 10 units away. Compute the volume of the water in the
 cup.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${32\pi }$}
Find the value of $k$ such that for a standardized normal variable $Z$, the probability $P(0 < Z < k)$ is equal to twice the probability $P(k < Z < 2k)$. Express your answer in terms of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.86}
. Decompose 2008 into a sum of natural numbers whose product is maximal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{668\times3+2\times2}
Consider the sequence: _, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3. What is the first number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A rectangle is divided by two straight lines into four smaller rectangles with perimeters of 6, 10, and 12 (the rectangle with a perimeter of 10 shares sides with the rectangles with perimeters of 6 and 12). Find the perimeter of the fourth rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
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{
The answer is 600.

}
Suppose a 2-fair carpool system is possible but a 1-fair carpool system is not. Must the number of MOPpers, \( n \), be even? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Given the operation \( \diamond \) defined as \( x\diamond 1 = x \) and \( x\diamond (n+1) = x^{x\diamond n} \) for \( n\in\mathbb{N} \), where \( x\in\mathbb{R}^+ \), let \( t \) be the unique positive real number such that \( t^t = 10^{10^{10}} \). It is known that \( 10^{10}>t \). If \( k \) and \( s \) are the unique positive real numbers satisfying \( k\diamond 50 = 10\diamond 51 \) and \( s\diamond 51 = 10\diamond 52 \), respectively, determine if \( k > s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k > s}
Let  $ a$  be a positive real numbers. In the coordinate plane denote by  $ S$  the area of the figure bounded by the curve  $ y=\sin x\ (0\leq x\leq \pi)$  and the  $ x$ -axis and denote  $ T$  by the area of the figure bounded by the curves  $ y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\　y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$  and the  $ x$ -axis. Find the value of  $ a$  such that  $ S: T=3: 1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ a = \frac{4}{3} }
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many 5-digit numbers does Kelvin like?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14034}
there are 20 cm , 21 cm , 22 cm , 23 cm , 24 cm , 25 cm rods of quantities 23 , 64 , 32 respectively . find average of this data .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.08}
Suppose that the edges of a 10-point graph can be colored with two colors in such a way that the graph does not contain a monochromatic triangle. What is the maximum number of edges the graph can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given the power series $A(x) = \sum_{n=0}^{\infty} a_nx^n$, where $A'(x) = 2xA(x)$, find a closed-form expression for $A(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_0e^{x^2}}
Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a sequence of nested intervals defined as follows: \( L_1 = [0,1] \). For each even \( n \), let \( L_n = [a_n, b_n] \) where \( a_n \) and \( b_n \) are irrational numbers such that \( a_n < b_n \) and \( a_n, b_n \in L_{n-1} \). For each odd \( n > 1 \), let \( L_n = [x_n, y_n] \) where \( x_n \) and \( y_n \) are rational numbers such that \( x_n < y_n \) and \( x_n, y_n \in L_{n-1} \). The sequence of intervals is such that \( L_1 \supset L_2 \supset L_3 \supset L_4 \supset L_5 \dots \). Define the set \( \mathcal{S} = \bigcap_{n=1}^{\infty} L_n \). Determine the number of elements in the set \( \mathcal{S} \) and whether these elements are rational, irrational, both, or neither.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\int_{\cos x}^{1}{}\arctan t ^2\ \mathrm dt}{\tan^2 x} \) using L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Determine the maximum number of faces of an n-sided pyramid that can be perpendicular to its base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-30}
Is the argument that the cardinality of \( V_\omega \) is \( \aleph_0 \) valid, given that each \( V_n \) in the von Neumann Hierarchy has finitely many elements and \( V_\omega = \bigcup_{n < \omega} V_n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}