inclusionAI/Ring-lite-rl-data huggingface dataset:
Avik is holding six identical ropes in his hand, with the midpoints in his fist. The ends of the ropes are lying on either side of his hand. Kamrul randomly selects and ties pairs of rope ends together on one side, and then does the same on the other side. If the probability of forming a loop with all six ropes is expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers, find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Point \( O \) divides segment \( AB \) into segments \( OA = 6 \) and \( OB = 4 \). A circle is drawn with center at point \( O \), and tangents to this circle are drawn from points \( A \) and \( B \), intersecting at point \( M \), with the points of tangency lying on the same side of line \( AB \). Find the radius of the circle if \( OM = 12 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6\sqrt{21}}{7}}
Given the function $g:\mathbb{R}^3 \to \mathbb{R}^3$ defined by $g(x, y, z) = (3x + 4z, 2x - 3z, x + 3y)$ and the region $S = \{(x, y, z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\}$, find the value of $\alpha$ for which the integral equality holds:

$$\iiint_{g(S)} (2x + y - 2z) \,dxdydz = \alpha \iiint_S z \,dxdydz$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. We need to arrange seating for 2 people, and these 2 people cannot sit next to each other (even one in the front and one in the back are considered not adjacent). Find the total number of different seating arrangements. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
What is the minimum number of colors required to paint all the cells of a 6 by 6 square grid such that each row, column, and diagonal of the square has all cells with different colors? Note: A diagonal of the square is defined as all rows of at least two cells going diagonally from one edge of the square to the other at an angle of $45^\circ$ or $135^\circ$ to the horizontal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
例7 设$a _ { i }$、$b _ { i } ( i = 1$,2,···,n)是有理数，使得对任意的实数x都有$x ^ { 2 } + x + 4 = \sum _ { i = 1 } ^ { n } ( a _ { i } x + b _ { i } ) ^ { 2 }$，求n的最小可能值．（2006年国家集训队试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The scoring rules for a soccer match are: 3 points for a win, 1 point for a draw, and 0 points for a loss. How many possible outcomes are there for a team that plays 14 matches and scores a total of 19 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.19"]}
Determine the limit points of the set \( S = \left\{ \left(m+\dfrac{1}{4^{|p|}},n+\dfrac{1}{4^{|q|}}\right) : m,n,p,q \in \mathbb{Z} \right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z} \times \mathbb{Z}}
In this figure, \( \angle RFS = \angle FDR \), \( FD = 4 \) inches, \( DR = 6 \) inches, \( FR = 5 \) inches, \( FS = 7\frac{1}{2} \) inches. Find the length of \( RS \), in inches, as a decimal. Multiply your final answer by 100 and provide the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}
The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
Find the radius of convergence of the power series $\sum_0^{\infty} n!x^{n^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the area of the figures bounded by the curves given in polar coordinates.

$$
r=2 \sin 4 \phi
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Mary picked 122 oranges, Jason picked 105 oranges, Tom picked 85 oranges, and Sarah picked 134 oranges. If there was a 10% increase in the number of oranges picked by each person, how many oranges were picked in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{491}
Mark rolls 5 fair 8-sided dice. What is the probability that at least three of the dice show the same number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1052}{8192}}
Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In the park, there are initially 25 oak trees, 40 maple trees, and 20 birch trees. Park workers will plant 73 oak trees, 52 maple trees, and 35 birch trees over the course of 3 days. On the first day, they will plant 40% of the oak trees, 50% of the maple trees, and 30% of the birch trees. On the second day, they will plant 35% of the oak trees, 25% of the maple trees, and 45% of the birch trees. On the third day, they will plant the remaining trees. Additionally, they will remove 15 oak trees, 10 maple trees, and 5 birch trees on the second day. Calculate the total number of oak, maple, and birch trees in the park after the workers have completed their tasks over the 3 days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{215}
There exist two triples of real numbers \((a,b,c)\) such that \(a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}\) are the roots to the cubic equation \(x^3-5x^2-15x+3\) listed in increasing order. Denote those \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\). If \(a_1, b_1,\) and \(c_1\) are the roots to the monic cubic polynomial \(f\) and \(a_2, b_2,\) and \(c_2\) are the roots to the monic cubic polynomial \(g\), find \(f(0)^3+g(0)^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-14}
Let $k$ be a real number. In the Cartesian coordinate plane $xOy$, consider the following sets of points:
\[ A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \]
\[ B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \]
Determine the value of $k$ such that the intersection of $A$ and $B$ is a single-element set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 - \sqrt{3}}
Compute the integral \( \int_{\gamma} \frac{f(z)}{z^3}dz \), where \( f(z) = az^3 + bz^2 + cz + d \) and \( \gamma : [0, 4\pi] \to \mathbb{C} \), \( \gamma(t) = e^{it} \). Use the Cauchy integral formula to find the result, considering the interval \([0, 4\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i b}
Find $b^2$ if the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{908}{49}}
The lateral edge of a regular quadrilateral pyramid makes an angle \( \alpha \) with the plane of the base. A plane is constructed through a vertex of the base and the midpoint of the opposite lateral edge, parallel to one of the diagonals of the base. Find the angle between this plane and the plane of the base of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan \left( \frac{\tan \alpha}{3} \right)}
On a circular track, Alphonse is at point \( A \) and Beryl is diametrically opposite at point \( B \). Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but different, speeds. After running for a while they notice that when they pass each other it is always at the same three places on the track. What is the ratio of their speeds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Let $f(x)$ be an even function defined on $\mathbb{R}$. For $x < 0$, $f(x) + x f'(x) < 0$, and $f(-4) = 0$. Determine the solution set of the inequality $x f(x) > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -4) \cup (4, \infty)}
A point \((X, Y)\) is uniformly distributed on the unit square \([0,1]^2\). Let \(\theta\) be the angle between the x-axis and the line segment connecting \((0,0)\) to the point \((X,Y)\). Find the expected value \(E[\theta]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Alexey plans to buy one of two car brands: "A" costing 900,000 rubles or "B" costing 600,000 rubles. On average, Alexey drives 15,000 km per year. The cost of gasoline is 40 rubles per liter. The cars use the same type of gasoline. The car is planned to be used for 5 years, after which the car of brand "A" can be sold for 500,000 rubles, and the car of brand "B" for 350,000 rubles.

| Car Brand | Fuel Consumption (L/100km) | Annual Insurance Cost (rubles) | Average Annual Maintenance Cost (rubles) |
| :--- | :--- | :--- | :--- |
| "A" | 9 | 35,000 | 25,000 |
| "B" | 10 | 32,000 | 20,000 |

Using the data from the table, answer the question: How much more expensive will it be for Alexey to buy and own the more expensive car?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160000}
A set of data arranged in ascending order is $1$, $4$, $4$, $x$, $7$, $8$ $(x \neq 7)$. If the median of this data set is $5$ times the mode, then the $60$th percentile of this data set is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
There is a method of setting a six-digit password: the first 2 digits are the last 2 digits of the birth year, the middle 2 digits are the birth month, and the last 2 digits are the birth date. For example, if someone was born on April 23, 1998, the six-digit password would be 980423. How many such possible passwords are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37,200}
Let \( f, g \in L^2 \). Find the limit \( \lim_{t \to \infty} \frac{\|f+tg\|_2-\|f\|_2}{t} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|g\|_2}
Evaluate the integral $$\int_{0}^{2\pi} \cos (m \theta)\cos(n \theta) \text{d} \theta$$ where $m$ and $n$ are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a game where a number $u$ is drawn from a uniform distribution on $[0,1]$. On one side of a piece of paper, $u$ is written, and on the other side, $u^2$ is written. One side of the paper is randomly drawn, and you win one dollar if you correctly guess whether the observed number is $u$ or $u^2$. Determine the expected value of this game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{8}}
Evaluate the inward flux of the vector field \( F = \langle y, -x, z \rangle \) over the surface \( S \) of the solid bounded by \( z = \sqrt{x^2+y^2} \) and \( z = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9\pi}
Calculate the total number of button presses required to try all possible 4-digit codes on a 10-key keyboard, assuming you do not need to press enter after each digit and consecutive digits can form different codes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10003}
Find the largest possible value of the product \( xyz \) for positive numbers \( x, y, \) and \( z \) such that the inequality \( \sqrt{\frac{x^2+y^2+z^2}{3}} \ge \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \) holds, and the answer is a two-digit number when simplified.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
What is the minimum number of points required to color an entire 3D space black using a device that colors all points at an irrational distance from each given point, assuming the initial color of the space is white?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the Jordan measure of the set \( M = \left\{ \begin{pmatrix} -r +2 t -t\\ 3r+t \\2r+2t \end{pmatrix} : 1\leq r \leq 3, 0 < s < 1, -1 < t \leq 2 \right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the real number \( a \) such that the conditional expectation \( E(X | Y = aX) = \frac{2}{9} \), given the joint probability density function for random variables \( X \) and \( Y \) is:
\[
f(x, y) = \begin{cases} 2(x+y) & \text{if } 0 < x < y < 1 \\ 0 & \text{otherwise} \end{cases}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the formulas for the surface area \( S_n(1) \) and volume \( V_n(1) \) of an \( n \)-sphere with radius 1:
\[ S_n(1) = \frac{n \pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)} \quad \text{and} \quad V_n(1) = \frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)} \]
where \( \Gamma(z) \) is the gamma function defined as:
\[ \Gamma(z) = \int_0^\infty e^{-x} x^{z-1}\, dx \]
Find the integers \( v \) and \( s \) such that \( S_s(1) \) and \( V_v(1) \) are maximum. Determine the value of \( v-s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the Fourier transform of the function \( xe^{-x^2/2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-i \sqrt{2\pi} \xi e^{-\xi^2/2}}
Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.1"]}
A closed cardboard box is in the shape of a cube. By cutting exactly 7 out of its 12 edges, the box can be unfolded onto a plane in various forms. How many distinct unfoldings can be obtained by cutting exactly 7 edges of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
250
}
Determine the cardinality of the set \( H = \{ f : \mathbb{N} \to \mathbb{N} \mid f \text{ is an involution} \} \), where an involution is a function that is its own inverse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
正方体旋转群元素最多的两个共轭类的元素个数之和为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the function \( f(n) = k \), where \( k \) is the \( n \)-th digit after the decimal point in the repeating decimal \( 0.\dot{9}1827364\dot{5} \), determine the value of \( \underbrace{f\{f \cdots f[f}_{1996 \text{ times }}(1)]\} \). Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A man's wages were decreased by 40%, then increased by 30%. After that, his wages were decreased by 20% and then increased by 10%. Calculate the overall percent change in his wages.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.36\%}
Given the surface parameterization \( g(s, t) = (s^2 + 2t, s + t, e^{st}) \), find \( \frac{\partial y}{\partial z} \) at the point \((3, e)\) near \( g(1, 1) = (3, 2, e) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3e}}
Calculate the second type integral
$$
\iint_{\sigma} x \, dy \, dz + y \, dx \, dz + z \, dx \, dy
$$
where $\sigma$ is the outer surface of the sphere $x^{2} + y^{2} + z^{2} = R^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \pi R^3}
Find the number of odd positive integers greater than 1 and less than 250 that are twinning, where an integer is twinning if for every prime divisor \( p \) of the integer, \( p - 2 \) divides the integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Solve the following equation:

$$
\left(5^{x}-2^{x-2}\right)^{2}+2 \log \left(5^{x}+2^{x-2}\right)=x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log_{\frac{2}{5}}4\approx-1.5129}
一只青蛙在正方形 $A B C D$ 的四个顶点间跳跃, 每次跳跃总是等可能地跳至与当前所在顶点相邻的两个顶点之一，且各次跳跃是独立的. 若青蛙第一次跳跃前位于顶点 $A$ ，则它第 6 次跳跃后恰好仍位于顶点 $A$ 的概率为 $\qquad$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{3}
Find the remainder when \(20^{25^{30^{35}}}\) is divided by 120.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Determine the asymptotic behavior of the expression:
\[ \frac{\int_{-n}^{n}ne^{-nx^{2}}dx}{\int_{-n}^{n}e^{-x^{2}}dx} \] as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n}}
How do I evaluate $$\int_\Gamma \frac{e^{iz}}{(z^2 + 1)^2}\,dz$$ where $\Gamma$ is the circle $|z| = 3$ positively oriented?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{e}\)}
Determine the number of linear Boolean functions from \({0, 1}^n\) to \({0, 1}\). A function \(F\) is linear if for all \(x, y \in {0, 1}^n\), \(F(x \oplus y) = F(x) \oplus F(y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Given the function $f(x)= \frac{1}{e} \cdot e^{x}+ \frac{a}{2}x^{2}-(a+1)x+a (a > 0)$, where $e$ is the base of the natural logarithm. If the function $y=f(x)$ and $y=f[f(x)]$ have the same range, find the maximum value of the real number $a$. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the double integral \( \iint_E x \, dx \, dy \) over the region \( E = \{(x,y) \mid 0 \le x, 0 \le y \le 1, 1 \le x^2 + y^2 \le 4\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$e^{\frac{1}{2}}$}
A pyramid's base is a square with unit side length. One of its side edges is also of unit length and coincides with the height of the pyramid. What is the largest dihedral angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
If the space vectors $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ satisfy $|\overrightarrow{{e_1}}|=|2\overrightarrow{{e_1}}+\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\overrightarrow{{e_1}}$ in the direction of $\overrightarrow{{e_2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3\sqrt{3}}{2}}
Given that \(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\) are 7 distinct prime numbers, and the sum of any two of \(a, b, c\) is 800. Let \(d\) be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1594}
There are 28 identical egg containers in the mart. Each egg container has the same number of eggs. When you select the 14th egg from the front, the 20th from the back, the 3rd from the left, and the 2nd from the right in an egg container, find all the number of eggs in the egg containers at this mart.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3696}
An engineer undertakes a project to build a road 15 km long in 300 days and employs 50 men for the purpose. After a certain number of days, he finds only 2.5 km of the road has been completed. He must employ approximately 75 extra men to finish the work in time. How many days had passed when he realized the progress?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
In triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to the internal angles $A$, $B$, $C$ respectively, and satisfy $\left(b-a\right)\left(\sin B+\sin A\right)=c(\sqrt{3}\sin B-\sin C)$. Find the value of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
In a round of the DES cryptosystem, the input is the 64-bit string $1010101010101010101010101010101010101010101010101010101010101010$, and the round key is the 48-bit string $111111111111111111111111111111111111111111111111$. Compute the 53rd bit of the output for this round.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number of distinct values obtained from the expression:

$$ \frac{n^2 - 2}{n^2 - n + 2} $$

where \( n \in \{1, 2, 3, \ldots, 100\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Three lattice points are chosen at random with replacement in the interior of the square defined by \(-99 \le x \le 100\) and \(-99 \le y \le 100\). The probability that the area of the triangle formed by these points is an integer 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{13}
Find the closure of \(l^1(\mathbb{N})\) in \(l^2(\mathbb{N})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{l^2(\mathbb{N})}
Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{\ln(1+nx)}{n x^n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
A circle passes through the vertices $A$ and $C$ of triangle $ABC$, intersects side $AB$ at point $D$ and side $BC$ at point $E$. Find the angle $CDB$ if $AD=5$, $AC=2\sqrt{7}$, $BE=4$, and $BD:CE=3:2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \left(-\frac{\sqrt{2}}{4}\right)}
Find the smallest integer \( n \) such that \( f_3(n) > T(n+3) \), using the bounds for \( f_3(n) \) provided.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Suppose that for a certain Poisson process, the average arrival rate is once every $4$ seconds. If there are no arrivals during the first $10$ seconds, what is the probability of observing exactly $6$ arrivals in the subsequent $10$ seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0278}
Determine whether the limit \( \lim_{t \to 0^{+}} A_t \) exists for the surface area \( A_t \) of the surface \( P_t = \{(x,y,z) : (x^2 + y^2)z = 1, \, t^2 \le x^2 + y^2 \le 1\} \) in \( \mathbb{R}^3 \). If it exists, evaluate the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the coefficient of $x^{10}$ in the generating function $$\frac{1+x^3}{1-2x+x^3-x^4}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{631}
Find the value of the infinite series: \[ \frac{1}{2\cdot4} + \frac{1\cdot3}{2\cdot4\cdot6} + \frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8} + \cdots \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}